On convertible nonnegative matrices * †

An n× nmatrix Ais called convertible if there is an n× n(1, -1)-matrix Hsuch that per A= det(H°A) where H ° Adenotes the Hadamard product of Hand A. A convertible (0,l)-matrix is called extremal if replacing any zero entry with a 1 breaks the convertibility. In this paper some properties of

nonnegative convertible matrices are investigated and some classes of extremal convertible (0,1)-matrices are obtained.     

Which compacta are noncommutative ARs?

We give a short answer to the question in the title: dendrits. Precisely we show that the C^*-algebra C(X) of all complex-valued continuous functions on a compactum X is projective in the category C¹ of all (not necessarily commutative) unital C^*-algebras if and only if X is an absolute retract of dimension dimX?1 or, equivalently, that X is a dendrit.     

Which algebraic groups are Picard varieties?

We show that every connected commutative algebraic group over an algebraically closed field of characteristic 0 is the Picard variety of some projective variety having only finitely many non-normal points.In contrast,no Witt group of dimension at least 3 over a perfect field of prime characteristic is isogenous to a Picard variety obtained by this construction.     

Which Real Lie Groups are Reductive?

The present author introduced in [4] the following version of reductivity: a real Lie group G with reductive Lie algebra g should be called reductive if its each irreducible continuous unitary representation is admissible. Furthermore, he showed there that his really reductive groups are reductive. Here we show the converse: each reductive real Lie group is really reductive, i.e., contains an open normal subgroup G1 of finite index such that G1 = Z(G1)G0, where G0 is the connected component of G containing 1, and Z(G1) is the center of G1.     

Which sets are images of disk-algebra functions?

We give a characterization of those compact sets in the plane with finitely many holes that are images of disk-algebra functions. We also show that the image of the closed unit disk via a polynomial is, in general, not polynomially convex.     

Which finite simple groups are unit groups?

We prove that if G$G$ is a finite simple group which is the unit group of a ring, then G$G$ is isomorphic to: (a) a cyclic group of order 2; or (b) a cyclic group of prime order 2^k−1$2^k-1$ for some k$k$; or (c) a projective special linear group _PSLn(_F2)${PSL}_n\left({\mathbb{F}}_2\right)$ for some n≥3$n\ge 3$. Moreover, these groups do all occur as unit groups. We deduce this classification from a more general result, which holds for groups G$G$ with no non-trivial normal 2-subgroup.     

Which distance-hereditary graphs are cover–incomparability graphs?

In this paper we deal with cover–incomparability graphs of posets, or briefly C–I graphs. These are graphs derived from posets as the edge-union of their cover graph and their incomparability graph. We answer two recently posed open questions. Which distance-hereditary graphs are C–I graphs? Which Ptolemaic (i.e. chordal distance-hereditary) graphs are C–I graphs? It follows that C–I graphs can be recognized efficiently in the class of all distance-hereditary graph whereas recognizing C–I graphs in general is known to be NP-complete.     

Which generalized Randi? indices are suitable measures of molecular branching?

Molecular branching is a very important notion, because it influences many physicochemical properties of chemical compounds. However, there is no consensus on how to measure branching. Nevertheless two requirements seem to be obvious: star is the most branched graph and path is the least branched graph. Every measure of branching should have these two graphs as extremal graphs. In this paper we restrict our attention to chemical trees (i.e. simple connected graphs with maximal degree at most 4), hence we have only one requirement that the path be an extremal graph. Here, we show that the generalized Randi? index R_p(G)=∑_uv∈E(G)(d_ud_v)^p is a suitable measure for branching if and only if p∈[λ,0)∪(0,λ^′) where λ is the solution of the equation in the interval (−0.793,−0.792) and λ^′ is the positive solution of the equation 3⋅3^x−2⋅2^x−4^x=0. These results include the solution of the problem proposed by Clark and Gutman.     

Which subnormal Toeplitz operators are either normal or analytic?

We study subnormal Toeplitz operators on the vector-valued Hardy space of the unit circle, along with an appropriate reformulation of P.R. Halmos?s Problem 5: Which subnormal block Toeplitz operators are either normal or analytic? We extend and prove Abrahamse?s theorem to the case of matrix-valued symbols; that is, we show that every subnormal block Toeplitz operator with bounded type symbol (i.e., a quotient of two bounded analytic functions), whose analytic and co-analytic parts have the “left coprime factorization”, is normal or analytic. We also prove that the left coprime factorization condition is essential. Finally, we examine a well-known conjecture, of whether every subnormal Toeplitz operator with finite rank self-commutator is normal or analytic.     

Which Faber–Moore–Chen digraphs are Cayley digraphs?

The problem of finding the largest graphs and digraphs of given degree and diameter is known as the ‘degree–diameter’ problem. One of the families of largest known vertex-transitive digraphs of given degree and diameter is the Faber–Moore–Chen digraphs. In our contribution we will classify those Faber–Moore–Chen digraphs that are Cayley digraphs.     

For Which Densities are Random Triangle-Free Graphs Almost Surely Bipartite?

Denote by the class of all triangle-free graphs on n vertices and m edges. Our main result is the following sharp threshold, which answers the question for which densities a typical triangle-free graph is bipartite. Fix > 0 and let . If n/2 m (1 – ) t ₃, then almost all graphs in are not bipartite, whereas if m (1 + )t ₃, then almost all of them are bipartite. For m (1 + )t ₃, this allows us to determine asymptotically the number of graphs in . We also obtain corresponding results for C -free graphs, for any cycle C of fixed odd length. Forschergruppe Algorithmen, Struktur, Zufall supported by Deutsche Forschungsgemeinschaft grant FOR 413/1-1     

A constructive version of the Boyle–Handelman theorem on the spectra of nonnegative matrices

A constructive version of the celebrated Boyle–Handelman theorem on the non-zero spectra of nonnegative matrices is presented.     

Semi Group Rings Which are Chinese Ring

In this paper we obtain conditions under which a semigroup ring is a Chinese ring.Further we define what are called weakly Chinese rings and study them.The authors in[1]called a commutative ring R to be a Chinese ring if,given elements a,b∈R and idealI,J(?)R such that a≡b(I J)there exists an element c∈R such that c≡a(I)andc≡b(J).For more properties about Chinese rings please refer[1].     

Large spaces of symmetric matrices of bounded rank are decomposable ∗

Let k and n be positive integers such that kn. Let S_n (F) denote the space of all n×n symmetric matrices over the field F with char F≠2. A subspace L of S_n (F) is said to be a k-subspace if rank Ak for every A?L.

Now suppose that k is even, and write k=2r. We say a k∥-subspace of S_n (F) is decomposable if there exists in Fⁿ a subspace W of dimension n?r such that x^tAx=0 for every x?W A?L.

We show here, under some mild assumptions on k n and F, that every k∥-subspace of S_n (F) of sufficiently large dimension must be decomposable. This is an analogue of a result obtained by Atkinson and Lloyd for corresponding subspaces of F^m,n .     

Two kinds of Pareto improvements of the economic system: An input–output analysis using the nonnegative matrices theory

This paper clearly reveals the economic meanings of final output rate, input multiplier, value-added rate, and output multiplier. Applying the output adjustment model and the price adjustment model as well as nonnegative matrices theory we find that if the matrix of intermediate output (or input) coefficients has at least one non-final (or non-initial) class then (i) an adjustment of output system enables the final output rates of whole or part sectors corresponding to the non-final classes to rise, the input multipliers of whole or part sectors corresponding to the non-final classes to decrease, and all other sectoral final output rates and input multipliers to be fixed, which is one kind of Pareto improvement of the economic system; (ii) the adaptation of price system enables the value-added rates of whole or part sectors corresponding to the non-initial classes to rise, the output multipliers of whole or part sectors corresponding to the non-initial classes to decrease, and all other sectoral value-added rates and output multipliers to be unchanged, which is another kind of Pareto improvement of the economic system. We respectively give a sufficient condition for the two kinds of Pareto improvements. A numerical example verifies these results.     

Which spaces for design?

We determine the largest class of spaces of sufficient regularity which are suitable for design in the sense that they do possess blossoms. It is the class of all spaces containing constants of which the spaces derived under differentiation are Quasi Extended Chebyshev spaces, i.e., they permit Hermite interpolation, Taylor interpolation excepted. It is also the class of all spaces which possess Bernstein bases, or of all spaces for which any associated spline space does possess a B-spline basis. Note that blossoms guarantee that such bases are normalised totally positive bases. They even are the optimal ones.     

Ando’s theorem for nonnegative forms

In this paper, we present a generalization of Ando??s theorem for nonnegative forms. He proved that the infimum of two positive operators A and B exists in the positive cone if and only if the generalized shorts [B]A and [A]B are comparable (see Ando et?al. in Problem of infimum in the positive cone, analytic and geometric inequalities and applications, Math. Appl. 478, pp 1?C12, 1999). That is, [A]B??? [B]A or [B]A??? [A]B. Using the concept of the parallel sum of nonnegative forms, Hassi, Sebestyén and de Snoo investigated the decomposability of a nonnegative form ${\mathfrak{t}}$ into an almost dominated and a singular part with respect to a nonnegative form ${\mathfrak{w}}$ (see Hassi et?al. in J. Funct. Anal. 257(12), 3858?C3894, 2009). Applying their results, we formulate a necessary and sufficient condition for the existence of the infimum of two nonnegative forms.