Retail competition and cooperative advertising

We consider a cooperative advertising channel consisting of a manufacturer selling its product through a retailer in competition with another independent retailer. The manufacturer subsidizes its retailer’s advertising only when a certain threshold is positive. Moreover, the manufacturer’s support for its retailer is higher under competition than in its absence.     

The role of store brand positioning for appropriating supply chain profit under shelf space allocation

We consider a retailer’s decision of developing a store brand (SB) version of a national brand (NB) and the role that its positioning strategy plays in appropriating the supply chain profit. Since the business of the retailer can be regarded as selling to NB manufacturers the shelf space at its disposal, we formulate a game-theoretical model of a single-retailer, single-manufacturer supply chain, where the retailer can decide whether to launch its own SB product and sells scarce shelf-space to a competing NB in a consumer good category. As a result, the most likely equilibrium outcome is that the available selling amount of each brand is constrained by the shelf-space available for its products and both brands coexist in the category. In this paper, we conceptualize the SB positioning that involves both product quality and product features. Our analysis shows that when the NB cross-price effect is not too large, the retailer should position its SB’s quality closer to the NB, more emphasize its SB’s differences in features facing a weaker NB, and less emphasize its SB’s differences in features facing a stronger NB. Our results stress the importance of SB positioning under the shelf-space allocation, in order to maximize the retailer’s value appropriation across the supply chain.     

The rational canonical form via the splitting field

It is clear that a given rational canonical form can be further resolved to a Jordan canonical form with entries from the splitting field of its minimal polynomial. Conversely, with an a priori knowledge of the existence and uniqueness of the rational canonical form of a matrix with entries from a general field, one can modify its Jordan canonical form in the splitting field of its minimal polynomial to construct its rational canonical form in the original field. No author has tried this converse with the a priori existence–uniqueness condition removed. It is feared that “in many occasions when, after a result has been established for a matrix with entries in a given field, considered as a matrix with entries in a finite extension of that field, we cannot go back from the extension field to get the desired information in the original field” [I.N. Herstein, Topics in Algebra, Ginn and Company, Waltham, MA, 1964 (pp. 262–263)]. The present paper removes this a priori condition and uses a “symmetrization” to “integrate” back the Jordan canonical form of a matrix to its rational canonical form in the original field.     

Perfect Semigroups and Aliens

Abstract. A topologized semigroup is called perfect if its multiplication is a perfect map (= a closed continuous mapping such that the inverse image of every point is compact). Thus a locally compact topological semigroup is perfect if and only if its multiplication is closed and each of its elements is compactly divided , that is, its divisors form a compact set. In the present paper we study compactly and non-compactly divided elements in the contexts of general locally compact semigroups, subsemigroups of groups, Lie semigroups and subsemigroups of Sl(2,R).     

On Integrable roots in split Lie triple systems

We focus on the notion of an integrable root in the framework of split Lie triple systems T with a coherent 0-root space. As a main result, it is shown that if T has all its nonzero roots integrable, then its standard embedding is a split Lie algebra having all its nonzero roots integrable. As a consequence, a local finiteness theorem for split Lie triple systems, saying that whenever all nonzero roots of T are integrable then T is locally finite, is stated. Finally, a classification theorem for split simple Lie triple systems having all its nonzero roots integrable is given.     

Perfect semigroups and aliens

A topologized semigroup is called perfect if its multiplication is a perfect map (= a closed continuous mapping such that the inverse image of every point is compact). Thus a locally compact topological semigroup is perfect if and only if its multiplication is closed and each of its elements is compactly divided, that is, its divisors form a compact set. In the present paper we study compactly and non-compactly divided elements in the contexts of general locally compact semigroups, subsemigroups of groups, Lie semigroups and subsemigroups of Sl(2, ?).     

Schemes of involving dual variables in inverse barrier functions for problems of linear and convex programming

With any stable map from a 3-manifold to ℝ³, we associate a graph with weights in its vertices and edges. These graphs are A-invariants from a global viewpoint. We study their properties and show that any tree with zero weights in its vertices and aleatory weights in its edges can be the graph of a stable map from S ³ to ℝ³.     

A generalized mean value property for polyharmonic functions

A well known property of a harmonic function in a ball is that its value at the centre equals the mean of its values on the boundary. Less well known is the more general property that its value at any point x equals the mean over all chords through x of its values at the ends of the chord, linearly interpolated at x. In this paper we show that a similar property holds for polyharmonic functions of any order when linear interpolation is replaced by two-point Hermite interpolation of odd degree.     

Integral representation for the solution of the stationary Schrödinger equation in a cone

In this paper, we consider the Dirichlet problem for the stationary Schrödinger equation in a cone with continuous boundary data. For a solution u of the stationary Schrödinger equation in a cone, we prove that if its positive part u⁺ satisfying a slowly growing condition, then its negative part u^? can also be dominated by a similar slowly growing condition. Meanwhile, u can be represented by its integral in the boundary of the cone.     

On convolutions of Siegel modular forms

In this article we study a Rankin‐Selberg convolution of n complex variables for pairs of degree n Siegel cusp forms. We establish its analytic continuation to ?ⁿ, determine its functional equations and find its singular curves. Also, we introduce and get similar results for a convolution of degree n Jacobi cusp forms. Furthermore, we show how the relation of a Siegel cusp form and its Fourier‐Jacobi coefficients is reflected in a particular relation connecting the two convolutions studied in this paper. As a consequence, the Dirichlet series introduced by Kalinin [7] and Yamazaki [19] are obtained as particular cases. As another application we generalize to any degree the estimate on the size of Fourier coefficients given in [14]. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Change in the properties of the membrane of the left ventricle of an artificial heart after operating in an animal

Conclusion After 53-day operation of a biomer membrane of an AH in a calf its physical and mechanical properties changed little. The membrane retained the greater part of its fatigue life and a high surface quality; as a consequence of insignificant creep, its functional properties also changed little.Translated from Mekhanika Kompozitnykh Materialov, No. 3, pp. 533–536, May–June, 1985.     

SPECTRAL PROPERTIES OF POSITIVE ELEMENTS IN BANACH LATTICE ALGEBRAS

Abstract

We prove that if a unital Banach lattice algebra has sufficiently many one-dimensional elements and if its unit element has sufficiently many components then its positive elements have spectral properties analogous to those of positive operators on Banach lattices. In particular, if a positive element is irreducible (in the sense that (1—e)xe > 0 for all components e of 1 satisfying 0 ≠ e ≠ 1) and compact, its spectral radius is positive and its spectrum shows cyclic behaviour.     

A criterion for the unique determination of domains in Euclidean spaces by the metrics of their boundaries induced by the intrinsic metrics of the domains

We say that a domain U ⊂ ℝ ⁿ is uniquely determined by the relative metric (which is the extension by continuity of the intrinsic metric of the domain on its boundary) of its Hausdorff boundary if any domain V ⊂ ℝ ⁿ such that its Hausdorff boundary is isometric in the relative metric to the Hausdorff boundary of U, is isometric to U in the Euclidean metric. In this paper, we obtain the necessary and sufficient conditions for the uniqueness of determination of a domain by the relative metric of its Hausdorff boundary.     

Parameśvara's rule for the circumradius of a cyclic quadrilateral

The expression for the circumradius of a cyclic quadrilateral in terms of its sides, usually attributed to L'Huilier in 1782, was known in India to Parame?vara (circa 1430). The present paper contains the original Sanskrit text of the rule, its English translation, and a discussion of its derivation as given by Sankara Váriar in his Kriy$\text{a}$kramakar$\text{i}$ (16th Century) along with relevant historical remarks.     

Edge-distance-regular graphs

Edge-distance-regularity is a concept recently introduced by the authors which is similar to that of distance-regularity, but now the graph is seen from each of its edges instead of from its vertices. More precisely, a graph Γ with adjacency matrix A is edge-distance-regular when it is distance-regular around each of its edges and with the same intersection numbers for any edge taken as a root. In this paper we study this concept, give some of its properties, such as the regularity of Γ, and derive some characterizations. In particular, it is shown that a graph is edge-distance-regular if and only if its k-incidence matrix is a polynomial of degree k in A multiplied by the (standard) incidence matrix. Also, the analogue of the spectral excess theorem for distance-regular graphs is proved, so giving a quasi-spectral characterization of edge-distance-regularity. Finally, it is shown that every nonbipartite graph which is both distance-regular and edge-distance-regular is a generalized odd graph.     

Random Walks on the Affine Group of a Homogeneous Tree in the Drift-Free Case

The affine group of a homogeneous tree is the group of all its isometries fixing an end of its boundary. We consider a random walk with law μ on this group and the associated random processes on the tree and its boundary. In the drift-free case there exists on the boundary of the tree a unique μ-invariant Radon measure. In this paper we describe its behaviour at infinity.     

Trees and circle orders

This paper continues a recent resurgence of interest in combinatorial properties of a poset that are associated with graph properties of its cover graph and order diagram. The following two theorems appearing in a 1977 paper of Trotter and Moore have played important roles in motivating this more modern research: (1) The dimension of a poset is at most 3 when its cover graph is at tree; (2) The dimension of a poset is at most 3 when the poset has a zero and its order diagram is planar. Although the underlying ideas lay dormant for more than 30 years, the first of these two results has become the base case for recent results bounding the dimension of a poset in terms of (a) the tree-width of its cover graph, and (b) the maximum dimension of its blocks. The second result is the base case for bounding the dimension of a planar poset in terms of the number of minimal elements. Continuing with this line of research, we show that every poset whose cover graph is a tree is a circle order, i.e., it has a representation as a family of circular disks in the Euclidean plane partially ordered by inclusion.     

Some geometric properties of Riemann's non-differentiable function

Riemann's non-differentiable function is a celebrated example of a continuous but almost nowhere differentiable function. There is strong numeric evidence that one of its complex versions represents a geometric trajectory in experiments related to the binormal flow or the vortex filament equation. In this setting, we analyse certain geometric properties of its image in $\mathbb{C}$. The objective of this note is to assert that the Hausdorff dimension of its image is no larger than 4/3 and that it has nowhere a tangent.