RESEARCH ANNOUNCEMENTS Odd Submodularity and Discrete Convexity

Lovasz asked in [1] whether any reasonable class of set-functions more generalthan submodular functions admits a similar deep theory. He pointed out that theminimum and the convolution of two submodular functions are not submodular ingeneral, but still have nice properties, such as integral vertices of the polyhedrainduced by them. We answer this question in this paper by three new set-functionclasses.     

Beiträge zur Theorie der geometrischen Objekte. VII

Ohne Zusammenfassung Vorgelegt von O. Varga     

Schrödinger Operators on Lattices. The Efimov Effect and Discrete Spectrum Asymptotics

The Hamiltonian of a system of three quantum mechanical particles moving on the three-dimensional lattice and interacting via zero-range attractive potentials is considered. For the two-particle energy operator h(k), with the two-particle quasi-momentum, the existence of a unique positive eigenvalue below the bottom of the continuous spectrum of h(k) for k 0 is proven, provided that h(0) has a zero energy resonance. The location of the essential and discrete spectra of the three-particle discrete Schrödinger operator H(K), being the three-particle quasi-momentum, is studied. The existence of infinitely many eigenvalues of H(0) is proven. It is found that for the number N(0, z) of eigenvalues of H(0) lying below the following limit exists with Moreover, for all sufficiently small nonzero values of the three-particle quasi-momentum K the finiteness of the number of eigenvalues of H(K) below the essential spectrum is established and the asymptotics for the number N(K, 0) of eigenvalues lying below zero is given. Communicated by Gian Michele GrafSubmitted 19/11/03, accepted 08/03/04     

Beiträge zur Arithmetik kommutativer Integritätsbereiche. VII. Inseparable Grundkörpererweiterung. Bemerkungen zur Körpertheorie

Ohne ZusammenfassungBeitrag VII schließt sich in der Fragestellung, in den benutzten Bezeichnungen usw. eng an Beitrag VI an [Math. Zeitschr.45 (1939), S. 1–19]. Auf die übrigen Beiträge wird nicht zurückgegriffen.     

A Note on the Pader "A Short Proof of Liapounoff''''s Convexity Theorem" of J. Lindenstrauss

J.Lindenstrauss once gave a short proof of Liapounoff＇s Convexity Theorem by using induction [1]. Now we give a more direct way to prove the theorem other than using induction. Here is the Liapounoff＇i theorem: Theorem Let μ_1,μ_2,…,μ_n,be finite positive non-atomic measures on some measure space X.Then M={(μ_1(A),μ_2(A),…,μ(A))|,measurable}is a closed and convex subset of R~n.     

Discrete–time nonlinear filtering with marked point process observations: ii. risk–sensitive filters

We discuss in this article the risk–sensitive filtering problem of estimating a nonlinear signal process, with nonadditive non–Gaussian noise, via a marked point process observation. This extends to the risk sensitive case all the risk–neutral results studied in Dufour and Kannan [2].By going into a change of measure, we derive the unnormalized conditional density of the signal conditioned on the observation history. We also derive the unnormalized prediction density. Using these, we present two separate expressions for the optimal estimate of the signal. A similar analysis of the smoothing density of the signal is also studied under both the risk–sensitive and risk–neutral cases. We specialize the above optimal estimation to the linear signal dynamics and marked point process observation under some Gaussian assumptions. We obtain a Kalman type risk-sensitive filter. Due to the special nature of the observation process, the conditional mean and covariance estimates directly depend now on the point process     

Discrete time queueing networks with product form steady state. Availability and performance analysis in an integrated model

For a discrete time network of generalized Bernoulli servers with unreliable nodes we derive the steady state probabilities for the joint queue length vector for all nodes and the availability status of the network. This allows us to assess the performance behavior and the reliability, resp. availability, of the network in an integrated model. Because our result exhibits a product form for the steady state distribution it opens the path to fast algorithmic evaluation of the desired performance and reliability indices.     

Corrigendum to “A criterion for the half-plane property” [Discrete Math. 309 (2008) 1385–1390]

We correct a small error in the paper mentioned in the title.     

Erratum to: “Local edge domination critical graphs” [Discrete Math. 161 (1996) 175–184]

The proof of the main theorem in the paper Henning et al. (1996) [2] is incorrect as it is missing an important case. Here we complete the proof by giving the missing case.     

A translation of the T. Levi-Civita paper “Interpretazione Gruppale degli Integrali di un Sistema Canonico” (Rend. Acc. Lincei, 1899, s. 3<Superscript>a</Superscript>, vol. VII,pp. 235–238)

In this paper we provide a translation of a paper by T. Levi-Civita, published in 1899, about the correspondence between symmetries and conservation laws for Hamilton’s equations. We discuss the results of this paper and their relationship with the more general classical results by E. Noether.     

Addendum to: “Hamilton-laceable bi-powers of locally finite bipartite graphs” [Discrete Math. 345 (2022) 112777]

In this addendum we give a short and easy negative answer to the two questions raised in the original article.     

Corrigendum to “The D-optimal saturated designs of order 22” [Discrete Math. 341 (2018) 380–387]

A matrix that was not included in Table 2 of the paper cited in the title is presented.