Rational invariants of even ternary forms under the orthogonal group

In this article we determine a generating set of rational invariants of minimal cardinality for the action of the orthogonal group $\mathrm{O}_3$ on the space $\mathbb{R}[x,y,z]_{2d}$ of ternary forms of even degree $2d$. The construction relies on two key ingredients: On one hand, the Slice Lemma allows us to reduce the problem to dermining the invariants for the action on a subspace of the finite subgroup $\mathrm{B}_3$ of signed permutations. On the other hand, our construction relies in a fundamental way on specific bases of harmonic polynomials. These bases provide maps with prescribed $\mathrm{B}_3$-equivariance properties. Our explicit construction of these bases should be relevant well beyond the scope of this paper. The expression of the $\mathrm{B}_3$-invariants can then be given in a compact form as the composition of two equivariant maps. Instead of providing (cumbersome) explicit expressions for the $\mathrm{O}_3$-invariants, we provide efficient algorithms for their evaluation and rewriting. We also use the constructed $\mathrm{B}_3$-invariants to determine the $\mathrm{O}_3$-orbit locus and provide an algorithm for the inverse problem of finding an element in $\mathbb{R}[x,y,z]_{2d}$ with prescribed values for its invariants. These are the computational issues relevant in brain imaging.Comment: v3 Changes: Reworked presentation of Neuroimaging application, refinement of Definition 3.1. To appear in "Foundations of Computational Mathematics

Human MLH1 Protein Participates in Genomic Damage Checkpoint Signaling in Response to DNA Interstrand Crosslinks, while MSH2 Functions in DNA Repair

DNA interstrand crosslinks (ICLs) are among the most toxic types of damage to a cell. For this reason, many ICL-inducing agents are effective therapeutic agents. For example, cisplatin and nitrogen mustards are used for treating cancer and psoralen plus UVA (PUVA) is useful for treating psoriasis. However, repair mechanisms for ICLs in the human genome are not clearly defined. Previously, we have shown that MSH2, the common subunit of the human MutSα and MutSβ mismatch recognition complexes, plays a role in the error-free repair of psoralen ICLs. We hypothesized that MLH1, the common subunit of human MutL complexes, is also involved in the cellular response to psoralen ICLs. Surprisingly, we instead found that MLH1-deficient human cells are more resistant to psoralen ICLs, in contrast to the sensitivity to these lesions displayed by MSH2-deficient cells. Apoptosis was not as efficiently induced by psoralen ICLs in MLH1-deficient cells as in MLH1-proficient cells as determined by caspase-3/7 activity and binding of annexin V. Strikingly, CHK2 phosphorylation was undetectable in MLH1-deficient cells, and phosphorylation of CHK1 was reduced after PUVA treatment, indicating that MLH1 is involved in signaling psoralen ICL-induced checkpoint activation. Psoralen ICLs can result in mutations near the crosslinked sites; however, MLH1 function was not required for the mutagenic repair of these lesions, and so its signaling function appears to have a role in maintaining genomic stability following exposure to ICL-induced DNA damage. Distinguishing the genetic status of MMR-deficient tumors as MSH2-deficient or MLH1-deficient is thus potentially important in predicting the efficacy of treatment with psoralen and perhaps with other ICL-inducing agents

The inverse EEG and MEG problems: The adjoint state approach I: The continuous case

In this report, we study the problem of the three-dimensional reconstruction of the electrical activity of the brain from electroencephalography (EEG) and magnetoencephalography (MEG). We use a variational approach based upon three main methods and ideas. The first one is the optimal control of systems governed by elliptic partial differential equations, the second is the regularization of the solutions while preserving the discontinuities (the edges), and the third one is the use of geometric information obtained from magnetic resonance images (MRI) to constrain the solutions in an anatomically "reasonable" way