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We study lattice sums \\(\\sum \\frac{1}{(\\|x\\|\\|y\\|\\|x+y\\|)^s}\\) taken over \\(SL_+(2,\\mathbb Z)\\), i.e.\\ the set of pairs \\((x,y)\\) of primitive lattice vectors in \\(\\mathbb Z_{\\geq 0}^2\\) with \\(\\det(x, y) = 1\\). We prove convergence of these and similar (determinant weighted) sums and introduce a new telescoping method on \\(SL_+(2,\\mathbb Z)\\) that yields, in particular, $$\\sum_{(x,y)\\in SL_+(2,\\mathbb Z)} \\frac{1}{\\|x\\|^2\\,\\|y\\|^2\\,\\|x+y\\|^2}=\\frac{\\pi}{4},$$ and a short proof of Zagier's identity \\(D_{1,1,1}=2E(z,3)+\\pi^3\\zeta(3)\\).

We study lattice sums \\(\\sum 1/(|x||y||x+y|)^s\\) taken over \\(SL_+(2,\\ZZ)\\), i.e. the set of pairs \\((x,y)\\) of primitive lattice vectors in \\(\\ZZ_{\\geq 0}^2\\) with \\(\\det(x, y) = 1\\). We prove convergence of these and similar (determinant weighted) sums and introduce a new telescoping method on \\(SL_+(2,\\ZZ)\\) that yields, in particular, $$\\sum_{(x,y)\\in SL_+(2,\\ZZ)} \\frac{1}{|x|^2|y|^2|x+y|^2}=\\frac{\\pi}{4},$$ and a short proof of Zagier's identity \\(D_{1,1,1}=2E(z,3)+\\pi^3\\zeta(3)\\).

These notes on creative telescoping are based on a series of lectures at the Institut Henri Poincare in November and December 2023.

We provide a powerful machinery to prove fully smooth one shot multipartite covering, aka convex split, type results for quantum states. In the important case of smooth multipartite convex split for classical quantum states, aka smooth multipartite soft covering, our machinery works even when certain marginals of these states do not satisfy pairwise independence. The recent paper (arXiv:2410.17893) gave the first proof of fully smooth multipartite convex split by simplifying and extending a technique called telescoping, developed originally for convex split by (arXiv:2304.12056). However, that work as well as all earlier works on convex split assumed pairwise or even more independence amongst suitable marginals of the quantum states. We develop our machinery by leveraging known results from (arXiv:1806.07278) involving tilting and augmentation smoothing of quantum states, combined with a novel observation that a natural quantum operation `flattening' quantum states actually preserves the fidelity. This machinery is powerful enough to lead to non pairwise independent results as mentioned above. As an application of our soft covering lemma without pairwise independence, we prove the `natural' one shot inner bounds for sending private classical information over a quantum wiretap interference channel, even when the classical encoders at the input lose pairwise independence in their encoding strategies to a certain extent. This result was unknown earlier even in the classical setting.

State-of-the-art Earth system models typically employ grid spacings of O(100 km), which is too coarse to explicitly resolve main drivers of the flow of energy and matter across the Earth system. In this paper, we present the new ICON-Sapphire model configuration, which targets a representation of the components of the Earth system and their interactions with a grid spacing of 10 km and finer. Through the use of selected simulation examples, we demonstrate that ICON-Sapphire can (i) be run coupled globally on seasonal timescales with a grid spacing of 5 km, on monthly timescales with a grid spacing of 2.5 km, and on daily timescales with a grid spacing of 1.25 km; (ii) resolve large eddies in the atmosphere using hectometer grid spacings on limited-area domains in atmosphere-only simulations; (iii) resolve submesoscale ocean eddies by using a global uniform grid of 1.25 km or a telescoping grid with the finest grid spacing at 530 m, the latter coupled to a uniform atmosphere; and (iv) simulate biogeochemistry in an ocean-only simulation integrated for 4 years at 10 km. Comparison of basic features of the climate system to observations reveals no obvious pitfalls, even though some observed aspects remain difficult to capture. The throughput of the coupled 5 km global simulation is 126 simulated days per day employing 21 % of the latest machine of the German Climate Computing Center. Extrapolating from these results, multi-decadal global simulations including interactive carbon are now possible, and short global simulations resolving large eddies in the atmosphere and submesoscale eddies in the ocean are within reach.

Several q-series identities were obtained using the telescoping method. These identities were then used to derive double series expressions for well-known zeta-value constants.

We provide a powerful machinery to prove fully smooth one shot multipartite covering, aka convex split, type results for quantum states. In the important case of smooth multipartite convex split for classical quantum states, aka smooth multipartite soft covering, our machinery works even when certain marginals of these states do not satisfy pairwise independence. The recent paper of Sen gave the first proof of fully smooth multipartite convex split by simplifying and extending a technique called telescoping, developed originally for convex split by (arXiv:2304.12056). However, that work as well as all earlier works on convex split assumed pairwise or even more independence amongst suitable marginals of the quantum states. We develop our machinery by leveraging known results from (arXiv:1806.07278) involving tilting and augmentation smoothing of quantum states, combined with a novel observation that a natural quantum operation `flattening' quantum states actually preserves the fidelity. This machinery is powerful enough to lead to non pairwise independent results as mentioned above. As an application of our soft covering lemma without pairwise independence, we prove the `natural' one shot inner bounds for sending private classical information over a quantum wiretap interference channel, even when the classical encoders at the input lose pairwise independence in their encoding strategies to a certain extent. This result was unknown earlier even in the classical setting.

Dynamic errors from the robotic machining process can negatively impact the accuracy of manufactured parts. Currently, effectively reducing dynamic errors in robotic machining remains a challenge due to the incomplete understanding of the relationship between machining parameters and dynamic errors, especially for hexapod machining cells. To address this topic, a dynamic error measurement strategy combining a telescoping ballbar, an unscented Kalman filter (UKF), and particle swarm optimization (PSO) was utilized in robotic machining. The machining parameters, including spindle speed, cutting depth, and feeding speed, were defined using the Taguchi method. Simultaneously, vibrations during machining were also systematically measured to fully comprehend the nature of dynamic errors. Experimental results indicate that dynamic errors in a hexapod machining cell (HMC) are significantly amplified in machining setups, ranging from 4 to 20 times greater compared to those of non-machining setups. These errors are particularly influenced by machining parameters, especially for spindle speed. Furthermore, the extracted dynamic errors exhibit comparable frequency distributions, such as spindle frequency and tool passing frequency, to the vibration signals obtained at the chosen sampling rate. This expands the application and enhances the comprehension of dynamic errors for spindle and cutting tool condition recognition.

Odontocete (echolocating whale) skulls exhibit extreme posterior displacement and overlapping of facial bones, here referred to as retrograde cranial telescoping. To examine retrograde cranial telescoping across 40 million years of whale evolution, we collected 3D scans of whale skulls spanning odontocete evolution. We used a sliding semilandmark morphometric approach with Procrustes superimposition and PCA to capture and describe the morphological variation present in the facial region, followed by Ancestral Character State Reconstruction (ACSR) and evolutionary model fitting on significant components to determine how retrograde cranial telescoping evolved. The first PC score explains the majority of variation associated with telescoping and reflects the posterior migration of the external nares and premaxilla alongside expansion of the maxilla and frontal. The earliest diverging fossil odontocetes were found to exhibit a lesser degree of cranial telescoping than later diverging but contemporary whale taxa. Major shifts in PC scores and centroid size are identified at the base of Odontoceti, and early burst and punctuated equilibrium models best fit the evolution of retrograde telescoping. This indicates that the Oligocene was a period of unusually high diversity and evolution in whale skull morphology, with little subsequent evolution in telescoping.

In this work, we describe the synthesis of new 4‐organyl‐5‐(organylselanyl)thiazol‐2‐amine hybrids through a one‐pot two‐step protocol. The transition metal‐free method involves the use of ultrasound as an alternative energy source and Oxone® as oxidant. To obtain the products, a telescoping approach was used, in which 4‐organylthiazol‐2‐amines were firstly prepared under ultrasonic irradiation, followed by the addition of diorganyl diselenides and Oxone®. Thus, 16 compounds were prepared, with yields ranging from 61 % to 98 %, using 2‐bromoacetophenone derivatives and diorganyl diselenides as easily available starting materials. A simple and efficient protocol promoted by ultrasonic irradiation to prepare new hybrids of 2‐amino‐4‐arylthiazoles and organoselenium was developed. The method involves two steps, in which 4‐organylthiazol‐2‐amine was first prepared under ultrasonic irradiation, followed by the addition of diorganyl diselenides and Oxone®. Thus, 16 compounds were obtained in yields from good to excellent (61–98 %) in short reaction times (30–40 min).