MYC locus rearrangements – often complex combinations of translocations, insertions, deletions, and inversions - in multiple myeloma (MM) were thought to be a late progression event, which often did not involve immunoglobulin genes. Yet germinal center activation of MYC expression has been reported to cause progression to MM in an MGUS prone mouse strain. Although previously detected in 16% of MM, we find MYC rearrangements in nearly 50% of MM, including smoldering MM, and they are heterogeneous in some cases. Rearrangements reposition MYC near a limited number of genes associated with conventional enhancers, but mostly with super-enhancers (e.g., IGH, IGL, IGK, NSMCE2, TXNDC5, FAM46C, FOXO3, IGJ, PRDM1). MYC rearrangements are associated with a significant increase of MYC expression that is monoallelic, but MM tumors lacking a rearrangement have bi-allelic MYC expression at significantly higher levels than in MGUS. We also show that germinal center activation of MYC does not cause MM in a mouse strain that rarely develops spontaneous MGUS. It appears that increased MYC expression at the MGUS/MM transition usually is bi-allelic, but sometimes can be mono-allelic if there is a MYC rearrangement. Our data suggests that MYC rearrangements, regardless of when they occur during MM pathogenesis, provide one event that contributes to tumor autonomy.

The paper reveals clear links between the differential-difference Kadomtsev-Petviashvili hierarchy and the (continuous) Kadomtsev-Petviashvili hierarchy, together with their symmetries, Hamiltonian structures and conserved quantities. They are connected through a uniform continuum limit. We derive isospectral and non-isospectral differential-difference Kadomtsev-Petviashvili flows through Lax triads, where the spatial variablex is looked as a new independent variable that is completely independent of the temporal variablet 1 . Such treatments not only enable us to derive the master symmetry as one of integrable non-isospectral flows, but also provide simple representations for both isospectral and nonisospectral differential-difference Kadomtsev-Petviashvili flows in terms of zero curvature equations. The obtained flows generate a Lie algebra with respect to Lie product ·, · , which further leads to two sets of symmetries for the isospectral differential-difference Kadomtsev-Petviashvili hierarchy, and the symmetries generate a Lie algebra, too. Making use of the recursive relations of the flows, symmetries and Noether operator we derive Hamiltonian structures for both isospectral and non-isospectral differential-difference Kadomtsev-Petviashvili hierarchies. The Hamiltonians generate a Lie algebra with respect to Poisson bracket {·, ·}. We then derive two sets of conserved quantities for the whole isospectral differential-difference Kadomtsev-Petviashvili hierarchy and they also generate a Lie algebra. All these obtained algebras have same basic structures. Then, we provide a continuum limit which is different from Miwa's transformation. By means of defining degrees of some elements with respect to the continuum limit, we prove that the differential-difference Kadomtsev-Petviashvili hierarchies together with their Lax triads, zero curvature representations and integrable properties go to their continuous counterparts in the continuum limit. Structure deformation of Lie algebras in the continuum limit is also explained.

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We present a method devised by Jacobi to derive Lagrangians of any second-order differential equation: it consists in finding a Jacobi Last Multiplier. We illustrate the easiness and the power of Jacobi's method by applying it to several equations, including a class of equations recently studied by Musielak with his own method [Z. E. Musielak, Standard and non-standard Lagrangians for dissipative dynamical systems with variable coefficients J. Phys. A: Math. Theor. 41 (2008) 055205], and in particular a Liènard type nonlinear oscillator and a second-order Riccati equation. Also, we derive more than one Lagrangian for each equation.

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Starting from integrable cellular automata we present a novel form of Painlevé equations. These equations are discrete in both the independent variable and the dependent one. We show that they capture the essence of the behavior of the Painlevé equations organizing themselves into a coalescence cascade and possessing special solutions. A necessary condition for the integrability of cellular automata is also presented.