Variants of the Two Machine Flow Shop Problem connected with factorization of matrix functions

In this paper we consider a number of variants of the Two Machine Flow Shop Problem. In these variants the makespan is given and the problem is to find a schedule that meets this makespan, thereby minimizing the infeasibilities of the jobs in a prescribed sense: In the max-variant the maximum infeasibility of the jobs is to be minimized, whereas in the sum-variant the objective is to minimize the sum of the infeasibilities of the jobs. For both variants observations about the structure of the optimal schedules are presented. In particular, it is proved that every instance of these problems has an optimal permutation schedule. It is also shown that the max-variant can be solved by Johnson's Rule. For the sum-variant this is not the case: For solving this problem to optimality something quite different is necessary. Both variants are connected with factorization problems for certain rational matrix functions. The factorizations involved are optimal in some sense and generalize the notion of complete factorization. In this way a connection is established between job scheduling theory on one hand, and mathematical systems theory on the other

Companion based matrix functions: description and minimal factorization

Companion based matrix functions are rational matrix functions admitting a minimal realization involving state space matrices that are first companions. Necessary and sufficient conditions are given for a rational matrix function to be companion based. Minimal factorization of such functions is discussed in detail. It is shown that the property of being companion based is hereditary with respect to minimal factorization. Also, the issue of minimal factorization is reduced to a division problem for pairs of monic polynomials of the same degree. In this context, a connection with the Euclidean algorithm is made. The results apply to canonical Wiener-Hopf factorization as well as to complete factorization. The analysis of the latter leads to a combinatorial problem involving the eigenvalues of the state space matrices. The algorithmic aspects of this problem are intimately related to the two machine flow shop problem and Johnson's rule from job scheduling theory

Logarithmic residues and sums of idempotents in the Banach algebra generated by the compact operators and the identity.

A logarithmic residue is a contour integral of the (left or right) logarithmic derivative of an analytic Banach algebra valued function. Logarithmic residues are intimately related to sums of idempotents. The present paper is concerned with logarithmic residues and sums of idempotents in the Banach algebra generated by the compact operators and the identity in the case when the underlying Banach space is infinite dimensional. The situation is more complex than encoutered in previous investigations. Logarithmic derivatives may have essential singularities and the geometric properties of the Banach space play a role. The set of sums of idempotens and the set of logarithmic residues have an intriguing topological structure

Logarithmic residues in Banach algebras

Let f be an analytic Banach algebra valued function and suppose that the contour integral of the logarithmic derivative f′f-1 around a Cauchy domain D vanishes. Does it follow that f takes invertible values on all of D? For important classes of Banach algebras, the answer is positive. In general, however, it is negative. The counterexample showing this involves a (nontrivial) zero sum of logarithmic residues (that are in fact idempotents). The analysis of such zero sums leads to results about the convex cone generated by the logarithmic residues

Logarithmic residues of analytic Banach algebra valued functions possessing a simply meromorphic inverse

A logarithmic residue is a contour integral of a logarithmic derivative (left or right) of an analytic Banach algebra valued function. For functions possessing a meromorphic inverse with simple poles only, the logarithmic residues are identified as the sums of idempotents. With the help of this observation, the issue of left versus right logarithmic residues is investigated, both for connected and nonconnected underlying Cauchy domains. Examples are given to elucidate the subject matter

Vector valued logarithmic residues and the extraction of elementary factors

An analysis is presented of the circumstances under which, by the extraction of elementary factors, an analytic Banach algebra valued function can be transformed into one taking invertible values only. Elementary factors are generalizations of the simple scalar expressions λ – α, the building blocks of scalar polynomials. In the Banach algebra situation they have the form e – p + (λ – α)p with p an idempotent. The analysis elucidates old results (such as on Fredholm operator valued functions) and yields new insights which are brought to bear on the study of vector-valued logarithmic residues. These are contour integrals of logarithmic derivatives of analytic Banach algebra valued functions. Examples illustrate the subject matter and show that new ground is covered. Also a long standing open problem is discussed from a fresh angle

Zero sums of idempotents in Banach algebras

The problem treated in this paper is the following. Let p1,..., pkbe idempotents in a Banach algebra B, and assume p1+...+pk=0. Does it follow that pj=0, j=1,..., k? For important classes of Banach algebras the answer turns out to be positive; in general, however, it is negative. A counterexample is given involving five nonzero bounded projections on infinite-dimensional separable Hilbert space. The number five is critical here: in Banach algebras nontrivial zero sums of four idempotents are impossible. In a purely algebraic context (no norm), the situation is different. There the critical number is four

Specific protein homeostatic functions of small heat-shock proteins increase lifespan

During aging, oxidized, misfolded, and aggregated proteins accumulate in cells, while the capacity to deal with protein damage declines severely. To cope with the toxicity of damaged proteins, cells rely on protein quality control networks, in particular proteins belonging to the family of heat-shock proteins (HSPs). As safeguards of the cellular proteome, HSPs assist in protein folding and prevent accumulation of damaged, misfolded proteins. Here, we compared the capacity of all Drosophila melanogaster small HSP family members for their ability to assist in refolding stress-denatured substrates and/or to prevent aggregation of disease-associated misfolded proteins. We identified CG14207 as a novel and potent small HSP member that exclusively assisted in HSP70-dependent refolding of stress-denatured proteins. Furthermore, we report that HSP67BC, which has no role in protein refolding, was the most effective small HSP preventing toxic protein aggregation in an HSP70-independent manner. Importantly, overexpression of both CG14207 and HSP67BC in Drosophila leads to a mild increase in lifespan, demonstrating that increased levels of functionally diverse small HSPs can promote longevity in vivo

The Copper Metabolism MURR1 Domain Protein 1 (COMMD1) Modulates the Aggregation of Misfolded Protein Species in a Client-Specific Manner

The Copper Metabolism MURR1 domain protein 1 (COMMD1) is a protein involved in multiple cellular pathways, including copper homeostasis, NF-kappa B and hypoxia signalling. Acting as a scaffold protein, COMMD1 mediates the levels, stability and proteolysis of its substrates (e.g. the copper-transporters ATP7B and ATP7A, RELA and HIF-1 alpha). Recently, we established an interaction between the Cu/Zn superoxide dismutase 1 (SOD1) and COMMD1, resulting in a decreased maturation and activation of SOD1. Mutations in SOD1, associated with the progressive neurodegenerative disorder Amyotrophic Lateral Sclerosis (ALS), cause misfolding and aggregation of the mutant SOD1 (mSOD1) protein. Here, we identify COMMD1 as a novel regulator of misfolded protein aggregation as it enhances the formation of mSOD1 aggregates upon binding. Interestingly, COMMD1 co-localizes to the sites of mSOD1 inclusions and forms high molecular weight complexes in the presence of mSOD1. The effect of COMMD1 on protein aggregation is client-specific as, in contrast to mSOD1, COMMD1 decreases the abundance of mutant Parkin inclusions, associated with Parkinson's disease. Aggregation of a polyglutamine-expanded Huntingtin, causative of Huntington's disease, appears unaltered by COMMD1. Altogether, this study offers new research directions to expand our current knowledge on the mechanisms underlying aggregation disease pathologies.</p

Stwl Modifies Chromatin Compaction and Is Required to Maintain DNA Integrity in the Presence of Perturbed DNA Replication

Hydroxyurea, a well-known DNA replication inhibitor, induces cell cycle arrest and intact checkpoint functions are required to survive DNA replication stress induced by this genotoxic agent. Perturbed DNA synthesis also results in elevated levels of DNA damage. It is unclear how organisms prevent accumulation of this type of DNA damage that coincides with hampered DNA synthesis. Here, we report the identification of stonewall ( stwl) as a novel hydroxyurea-hypersensitive mutant. We demonstrate that Stwl is required to prevent accumulation of DNA damage induced by hydroxyurea; yet, Stwl is not involved in S/M checkpoint regulation. We show that Stwl is a heterochromatin-associated protein with transcription-repressing capacities. In stwl mutants, levels of trimethylated H3K27 and H3K9 ( two hallmarks of silent chromatin) are decreased. Our data provide evidence for a Stwl-dependent epigenetic mechanism that is involved in the maintenance of the normal balance between euchromatin and heterochromatin and that is required to prevent accumulation of DNA damage in the presence of DNA replication stress.</p