Infinite families of solutions for A3+B3=C3+D3A^3 + B^3 = C^3 + D^3 and A4+B4+C4+D4+E4=F4A^4 + B^4 + C^4 + D^4 + E^4 = F^4

Ramanujan, in his lost notebook, gave an interesting identity, which generates infinite families of solutions to Euler's Diophantine equation $A^3 + B^3 = C^3 + D^3$. In this paper, we produce a few infinite families of solutions to the aforementioned Diophantine equation as well as for the Diophantine equation $A^4 + B^4 + C^4 + D^4 + E^4 = F^4$ in the spirit of Ramanujan.Comment: 16 page

Mosaic Variant of Turner Syndrome

Turner syndrome is the most common chromosomal abnormality leading to gonadal failure and primary amenorrhea. While half of the cases have monosomy of chromosome X, the remaining exhibit mosaicism resulting in wide variation of phenotypic characteristics and clinical manifestations. We present a case of a 24-year-old female with mosaic variant Turner syndrome. The diagnosis was confirmed by karyotype analysis and laparoscopy

Open X-Embodiment:Robotic learning datasets and RT-X models

Large, high-capacity models trained on diverse datasets have shown remarkable successes on efficiently tackling downstream applications. In domains from NLP to Computer Vision, this has led to a consolidation of pretrained models, with general pretrained backbones serving as a starting point for many applications. Can such a consolidation happen in robotics? Conventionally, robotic learning methods train a separate model for every application, every robot, and even every environment. Can we instead train "generalist" X-robot policy that can be adapted efficiently to new robots, tasks, and environments? In this paper, we provide datasets in standardized data formats and models to make it possible to explore this possibility in the context of robotic manipulation, alongside experimental results that provide an example of effective X-robot policies. We assemble a dataset from 22 different robots collected through a collaboration between 21 institutions, demonstrating 527 skills (160266 tasks). We show that a high-capacity model trained on this data, which we call RT-X, exhibits positive transfer and improves the capabilities of multiple robots by leveraging experience from other platforms. The project website is robotics-transformer-x.github.io

Biophysical and structural characterization of the interaction of the circadian transcription factor BMAL1 with its coactivator CBP

The mammalian CLOCK:BMAL1 transcription factor complex and its coactivators CREB-binding protein (CBP)/p300 and mixed-lineage leukemia 1 (MLL1) play a central role in circadian transcriptional regulation and chromatin modification. The interactions of BMAL1’s C-terminal transactivation domain (TAD) with the KIX domain of CBP/p300 (activating) and with CRY1 (repressing) as well as the BMAL1 G-region preceding the TAD regulate the circadian oscillations. The repressive BMAL1-TAD-CRY1 interactions are enhanced by the circadian acetylation of Lys-537 within the BMAL1 G-region. The CBP-KIX domain interacts with a plethora of transcription regulators via its two distinct pockets referred to as MLL- and CREB-pKID/c-Myb-binding pockets, typically targeting the intrinsically disordered regions within the TAD, which often attain folding upon binding to the KIX domain. In this thesis, we characterized the interaction of the CBP-KIX domain with BMAL1 proteins including the BMAL1-TAD, parts of the G-region, and Lys-537. Tethering the small compound 1-10 in the MLL-binding pocket of the CBP-KIX domain weakened BMAL1 binding and MLL1-bound KIX did not form a ternary complex with BMAL1, indicating that the MLL-binding pocket is important for KIX-BMAL1 interactions. Additionally, mutations in the second pKID/c-Myb-binding pocket of the KIX domain moderately impacted BMAL1 binding. The BMAL1(K537Q) mutation mimicking Lys-537 acetylation, however, did not affect the KIX-binding affinity, in contrast to its enhancing effect on CRY1 binding. Moreover, KIX binding does not induce the formation of extended regular secondary structures in BMAL1. SAXS models of BMAL1 and BMAL1:KIX complexes revealed that the N-terminal BMAL1 G-region including Lys-537 forms elongated extensions emerging from the bulkier BMAL1-TAD:KIX core complex. Fitting high-resolution KIX domain structures into the SAXS-derived envelopes suggested that the G-region emerges near the MLL-binding pocket, further supporting a role of this pocket in BMAL1 binding. This study significantly advances the mechanistic understanding of the roles of the BMAL1-TAD-CBP-KIX interaction and its interplay with other KIX ligands and with the BMAL1-CRY1 interaction in circadian gene regulation

Riesz-type criteria for the Riemann hypothesis

In 1916, Riesz proved that the Riemann Hypothesis is equivalent to the bound ∑ n = 1 ∞ μ ( n ) n 2 exp ⁡ ( − x n 2 ) = O ϵ ( x − 3 4 + ϵ ) \sum _{n=1}^\infty \frac {\mu (n)}{n^2} \exp \left ( - \frac {x}{n^2} \right ) = O_{\epsilon } \left ( x^{-\frac {3}{4} + \epsilon } \right ) , as x → ∞ x \rightarrow \infty , for any ϵ &gt; 0 \epsilon &gt;0 . Around the same time, Hardy and Littlewood gave another equivalent criterion for the Riemann Hypothesis while correcting an identity of Ramanujan. In the present paper, we establish a one-variable generalization of the identity of Hardy and Littlewood and as an application, we provide Riesz-type criteria for the Riemann Hypothesis. In particular, we obtain the bound given by Riesz as well as the bound of Hardy and Littlewood.</p

Filament assembly underpins the double-stranded DNA specificity of AIM2-like receptors

AbstractUpon sensing cytosolic- and/or viral double-stranded (ds)DNA, absent-in-melanoma-2 (AIM2)-like-receptors (ALRs) assemble into filamentous signaling platforms to initiate inflammatory responses. The versatile yet critical roles of ALRs in host innate defense are increasingly appreciated; however, the mechanisms by which AIM2 and its related IFI16 specifically recognize dsDNA over other nucleic acids remain poorly understood (i.e. single-stranded (ss)DNA, dsRNA, ssRNA and DNA:RNA hybrid). Here, we find that although AIM2 can interact with various nucleic acids, it preferentially binds to and assembles filaments faster on dsDNA in a duplex length-dependent manner. Moreover, AIM2 oligomers assembled on nucleic acids other than dsDNA not only display less ordered filamentous structures, but also fail to induce the polymerization of downstream ASC. Likewise, although showing broader nucleic acid selectivity than AIM2, IFI16 binds to and oligomerizes most readily on dsDNA in a duplex length-dependent manner. Nevertheless, IFI16 fails to form filaments on single-stranded nucleic acids and does not accelerate the polymerization of ASC regardless of bound nucleic acids. Together, we reveal that filament assembly is integral to nucleic acid distinction by ALRs.</jats:p