A note on addition chains

New proofs are given for certain cases of the Scholz-Brauer problem in addition chains.     

A note on addition theorems for Mathieu functions

Zusammenfassung Es wird gezeigt, dass ein Additionstheorem für Mathieusche Funktionen-verschieden von dem vonMeixner undSchäfke gegebenen-existiert.     

A note on the core

This note studies an exchange economy in which there are n traders and n “kinds” of commodities. Each trader has n utility functions corresponding to n “kinds” of commodities, respectively. Thus, a multiple non-transferable utility game can be derived from this exchange economy. It is shown that a sufficient condition for non-emptiness of the core of a multiple non-transferable utility game. The result is an extension of Scarf-Billera theorem.     

A note on the nucleolus

It was shown byKohlberg [1972] that the nucleolus can be obtained by solving a linear program of extremely large size (2 ⁿ ! constraints). We show here how this program can be reduced to a more tractable size (4 ⁿ constraints).     

A note on the Newton radius

The Newton radius of a code is the largest weight of a uniquely correctable error. We establish a lower bound for the Newton radius in terms of the rate. In particular we show that in any family of linear codes of rate below one half, the Newton radius increases linearly with the codeword length.     

A note on the Jacobian Conjecture

In this note, we show that, if the Druzkowski mappings F(X)=X+(AX)^∗3, i.e. F(X)=(x₁+(a₁₁x₁+?+a_1nx_n)³,…,x_n+(a_n1x₁+?+a_nnx_n)³), satisfies TrJ((AX)^∗3)=0, then where δ is the number of diagonal elements of A which are equal to zero. Furthermore, we show the Jacobian Conjecture is true for the Druzkowski mappings in dimension ?9 in the case .     

A note on the classification ofUHF-algebras

In this paper we study the effect of connectivity and simple connectivity of the group of invertible elements on the type of aUHF-algebra.     

A note on the MIR closure

In 1988, Nemhauser and Wolsey introduced the concept of MIR inequality for mixed integer linear programs. In 1998, Wolsey gave another definition of MIR inequalities. This note points out that the natural concepts of MIR closures derived from these two definitions are distinct. Dash, Günlük and Lodi made the same observation independently.     

A note on the dichotomy spectrum

We point out an error in the above short note and correct it under additional assumptions.     

A note on the Gelfand-Mazur Theorem

Three Gelfand-Mazur type theorems are proved. One of these provides a -property analogue of Zalar's recent generalizations of the Froelich-Ingelstam-Smiley Theorems concerning unital multiplication in Hilbert spaces; the second illustrates that the assumption in Kaplansky's version of the Gelfand-Mazur Theorem can be weakened in the presence of a -norm; whereas the third provides a real analogue of a result due to Srinivasan.

     

A note on the DiPerna-Lions Flows

In this note, we give a short proof for the DiPerna-Lions flows associated to ODEs following the method of Crippa and De Lellis [3]. More precisely, assume that [divb] ∈ Ll∞oc(Rd), |b|/(1 + |x| log |x|) ∈ L∞(Rd) and | b| φ(| b|) ∈ Ll1oc(Rd), where φ(r) = log log(r + c), c > 0. Then, there exists a unique regular Lagrangian flow associated with the ODE X˙(t, x) = b(X(t, x)), X(0, x) = x.     

A note on the shameful conjecture

Let $P_G\left(q\right)$ denote the chromatic polynomial of a graph $G$ on $n$ vertices. The ‘shameful conjecture’ due to Bartels and Welsh states that, $\frac{P_G\left(n\right)}{P_G(n-1)}\ge \frac{n^n}{{(n-1)}^n}\text{.}$ Let $\mu \left(G\right)$ denote the expected number of colors used in a uniformly random proper $n$-coloring of $G$. The above inequality can be interpreted as saying that $\mu \left(G\right)\ge \mu \left(O_n\right)$, where $O_n$ is the empty graph on $n$ nodes. This conjecture was proved by F.M. Dong, who in fact showed that, $\frac{P_G\left(q\right)}{P_G(q-1)}\ge \frac{q^n}{{(q-1)}^n}$ for all $q\ge n$. There are examples showing that this inequality is not true for all $q\ge 2$. In this paper, we show that the above inequality holds for all $q\ge 36D^{3/2}$, where $D$ is the largest degree of $G$. It is also shown that the above inequality holds true for all $q\ge 2$ when $G$ is a claw-free graph.