Shuffle product of desingularized multiple zeta functions at integer points

We provide an estimate for the number of nontrivial integer points on the Pellian surface t 2 - d u 2 = 1 in a bounded region. We give a lower bound on the size of fundamental solutions for almost all d in a certain class, based on a recent conjecture of Browning and Wilsch about integer points on log K3 surfaces. We also obtain an upper bound on the average of class number in this class, assuming the same conjecture.

This paper constitutes a step in the direction of developing integer lattice gas methods as an attractive alternative to lattice Boltzmann methods. Here we show that to Boltzmann limit the one-dimensional Blommel integer lattice gas is very close to entropic lattice Boltzmann. More interestingly, the integer lattice gas retains additional correlations that prevent the existence of a well-defined Boltzmann limit. In the analysis of the decaying sine wave we will see that in some situations the bulk viscosity can crucially depend on such correlations beyond the Boltzmann limit. A sampling collision operator, introduced here, can speed up the execution time to make the algorithm obtain comparable computational efficiency to entropic lattice Boltzmann methods.

A Novel Clinical Risk Score to Predict Vasoplegia After Adult Cardiac Surgery.

The smoothing parameter on lattices is crucial for lattice-based cryptographic design. In this study, we establish a new upper bound for the lattice smoothing parameter, which represents an improvement over several significant classical findings. For one-dimensional integer lattices, under specific and optimized conditions, we have achieved a more precise upper bound compared to previous research. Regarding general high-dimensional lattices, when the lattice dimension is large enough and the error parameter is within a particular range, we have derived a new upper bound. In the practical applications of lattice-based cryptography, where the lattice dimension is typically large, our new bound enables a more natural and smaller setting for the error parameter, thereby improving the upper bounds on all known smoothing parameters.

Building upon the Integer Lattice Gas Automata framework of Blommel and Wagner [Phys. Rev. E97, 023310 (2018)2470-004510.1103/PhysRevE.97.023310], we introduce a simplified, fluctuation-free variant. This approach relies on floating-point numbers and closely mirrors the Lattice Boltzmann Method (LBM), with the key distinction being a different collision operator. This operator, derived from the ensemble average of transition probabilities, generates nonlinear terms. We propose this float lattice gas automata collision as a computationally efficient alternative to traditional and quantum LBM implementations.

As observed in the case of COVID-19, effective vaccines for an emerging pandemic tend to be in limited supply initially and must be allocated strategically. The allocation of vaccines can be modeled as a discrete optimization problem that prior research has shown to be computationally difficult (i.e., NP-hard) to solve even approximately. Using a combination of theoretical and experimental results, we show that this hardness result may be circumvented. We present our results in the context of a metapopulation model, which views a population as composed of geographically dispersed heterogeneous subpopulations, with arbitrary travel patterns between them. In this setting, vaccine bundles are allocated at a subpopulation level, and so the vaccine allocation problem can be formulated as a problem of maximizing an integer lattice function subject to a budget constraint . We consider a variety of simple, well-known greedy algorithms for this problem and show the effectiveness of these algorithms for three problem instances at different scales: New Hampshire (10 counties, population 1.4 million), Iowa (99 counties, population 3.2 million), and Texas (254 counties, population 30.03 million). We provide a theoretical explanation for this effectiveness by showing that the approximation factor (a measure of how well the algorithmic output for a problem instance compares to its theoretical optimum) of these algorithms depends on the submodularity ratio of the objective function g. The submodularity ratio of a function is a measure of how distant g is from being submodular; here submodularity refers to the very useful “diminishing returns” property of set and lattice functions, i.e., the property that as the function inputs are increased the function value increases, but not by as much.

The search rank-2 module Lattice Isomorphism Problem (smLIP), over a cyclotomic ring of degree a power of two, can be reduced to an instance of the Lattice Isomorphism Problem (LIP) of at most half the rank if an adversary knows a nontrivial automorphism of the underlying integer lattice. Knowledge of such a nontrivial automorphism speeds up the key recovery attack on HAWK at least quadratically, which would halve the number of security bits. Luo et al. (ASIACRYPT 2024) recently found an automorphism that breaks omSVP, the initial underlying hardness assumption of HAWK. The team of HAWK amended the definition of omSVP to include this so-called symplectic automorphism in their submission to the second round of NIST's standardization of additional signatures. This work provides confidence in the soundness of this updated definition, assuming smLIP is hard, since there are plausibly no more trivial automorphisms that allow winning the omSVP game easily. Although this work does not affect the security of HAWK, it opens up a new attack avenue involving the automorphism group that may be theoretically interesting on its own.

We introduce an algebra of elliptic commuting variables involving a base $q$, nome $p$, and $2r$ noncommuting variables. This algebra, which for $r=1$ reduces to an algebra considered earlier by the author, is an elliptic extension of the well-known algebra of $r$ $q$-commuting variables. We present a multinomial theorem valid as an identity in this algebra, hereby extending the author's previously obtained elliptic binomial theorem to higher rank. Two essential ingredients are a consistency relation satisfied by the elliptic weights and the Weierstraß type $\mathsf A$ elliptic partial fraction decomposition. From the elliptic multinomial theorem we obtain, by convolution, an identity equivalent to Rosengren's type $\mathsf A$ extension of the Frenkel-Turaev ${}_{10}V_9$ summation. Interpreted in terms of a weighted counting of lattice paths in the integer lattice $\mathbb Z^r$, this derivation of Rosengren's $\mathsf A_r$ Frenkel-Turaev summation constitutes the first combinatorial proof of that fundamental identity.

Summary This paper treats triangles in the plane whose vertices lie on the integer lattice; i.e., the vertices have integer coordinates. It shows that, apart from trivial examples, the circumcenter, centroid, and orthocenter of such triangles never all lie on the integer lattice. Several further observations are made concerning the circumcenter, centroid, and orthocenter.

A numerical semigroup is a cofinite subset of ℤ ≥0 containing 0 and closed under addition. Each numerical semigroup S with smallest positive element m corresponds to an integer point in the Kunz cone 𝒞 m ⊆ℝ m-1 , and the face of 𝒞 m containing that integer point determines certain algebraic properties of S. In this paper, we introduce the Kunz fan, a pure, polyhedral cone complex comprised of a faithful projection of certain faces of 𝒞 m . We characterize several aspects of the Kunz fan in terms of the combinatorics of Kunz nilsemigroups, which are known to index the faces of 𝒞 m , and our results culminate in a method of “walking” the face lattice of the Kunz cone in a manner analogous to that of a Gröbner walk. We apply our results in several contexts, including a wealth of computational data obtained from the aforementioned “walks” and a proof of a recent conjecture concerning which numerical semigroups achieve the highest minimal presentation cardinality when one fixes the smallest positive element and the number of generators.

Consider a single auctioneer who wants to sell multiple units of distinct indivisible items to bidders with private valuations. The set of feasible allocations is constrained to integer base points, which are the integer points of an integer base polyhedron. Each bidder’s valuation is the integer restriction of a sum of nondecreasing, concave single-parameter functions. This seemingly abstract setting is of theoretical relevance and has various interesting applications. In this context, we develop an ascending auction that implements a social welfare-maximizing allocation, charges Vickrey–Clarke–Groves prices, relies only on a single price, is ex-post incentive-compatible, and satisfies unconditional winner privacy. The auction has a polynomial running time in the number of bidders, items, and units; in the case of linear separable valuations it runs even in strongly polynomial time, thereby improving on the literature. Moreover, by relaxing unconditional winner privacy, the auction can be made fully polynomial in the number of bidders, items, units, and integer breakpoints of bidders’ valuations. If we assume that bidders are unit-demand , then our auction is dominant-strategy incentive-compatible and (weakly) group strategy-proof, much like deferred acceptance auctions.

The results of strength and weight efficiency analysis of composite fuselage sections with lattice structure layouts are presented. A rational nonregular layout of the lattice grid is proposed, which has an enhanced weight efficiency in comparison with the regular (periodic, along the length of the fuselage) one. The rational non-regular grid layout is formed by changing the geodesic direction of the spiral ribs with a constant angle of orientation to the longitudinal axis of the fuselage for a curvilinear direction of the spiral ribs. The validation of the proposed lattice grid layout was carried out using an automated parametric method for FEM strength analysis of lattice composite barrels. As objects of investigation, a number of barrels of an elongated fuselage structure of a perspective passenger aircraft were selected, having a diameter of 4m and lengths from 6 to 10 m. The weight effectiveness of the proposed layout was shown.

Sparse data structures are ubiquitous in modern computing, and numerous formats have been designed to represent them. These formats may exploit specific sparsity patterns, aiming to achieve higher performance for key numerical computations than more general-purpose formats such as CSR and COO. In this work we present UZP, a new sparse format based on polyhedral sets of integer points. UZP is a flexible format that subsumes CSR, COO, DIA, BCSR, etc., by raising them to a common mathematical abstraction: a union of integer polyhedra, each intersected with an affine lattice. We present a modular approach to building and optimizing UZP: it captures equivalence classes for the sparse structure, enabling the tuning of the representation for target-specific and application-specific performance considerations. UZP is built from any input sparse structure using integer coordinates, and is interoperable with existing software using CSR and COO data layout. We provide detailed performance evaluation of UZP on 200+ matrices from SuiteSparse, demonstrating how simple and mostly unoptimized generic executors for UZP can already achieve solid performance by exploiting Z-polyhedra structures.

When solving systems of polynomial equations and inequalities, the task of computing their solutions with integer coordinates is a much harder problem than that of computing their real solutions or that of computing all their solutions. In fact, in the presence of non-linear constraints this task may simply become an undecidable problem. However, studying the integer solutions of systems of equations and inequalities is of practical importance in areas like combinatorial optimization and compiler construction. Over the past 10 years, a series of projects has equipped the computer algebra system Maple with a number of tools for dealing with the integer points of systems of linear equations and inequalities, even in the presence of parameters. With these tools, one can either decide whether integer point solutions exist, or count them, or describe them in a compact way, or enumerate them. In this paper, we will give a tour of these facilities and illustrate their usage with a number of applications. In this paper, we have discussed the adaptation of Barvinok’s algorithm for integer point counting from non-parametric polytopes to parametric polytopes. We report on a Maple implementation of this adaptation. This includes a general framework called Value- UnderConstraints in support of parametric system solving. Additionally, in order to represent the parametric results of these counting of para- metric system, a new Maple data structure called Quasi-polynomial was built.

We consider kernel operators defined by a dynamical system. The Hausdorff distance of spectra is estimated by the Hausdorff distance of subsystems. We prove that the spectrum map is 1/2 -Hölder continuous provided the group action and kernel are Lipschitz continuous and the group has strict polynomial growth. Also, we prove that the continuity can be improved resulting in the spectrum map being Lipschitz continuous provided the kernel is instead locally-constant. This complements a result by J. Avron, P. van Mouche, and B. Simon (1990) establishing that one-dimensional discrete quasiperiodic Schrödinger operators with Lipschitz continuous potentials, e.g., the almost-Mathieu operator, exhibit spectral 1/2 -Hölder continuity. Also, this complements a result by S. Beckus, J. Bellissard, and H. Cornean (2019) establishing that d -dimensional discrete subshift Schrödinger operators with locally-constant potentials, e.g., the Fibonacci Hamiltonian, exhibit spectral Lipschitz continuity. Our work exposes the connection between the past two results, and the group, e.g., the Heisenberg group, needs not be the integer lattice nor abelian.

Patients who had venous thromboembolism (VTE) are not only at increased risk of recurrent VTE but also of major adverse cardiovascular events (MACEs) than the general population. Therefore, the prediction of the risk of these events is important for a tailored prevention and mitigation strategy. We aimed to develop simple scores to estimate recurrent VTE and MACE risks after the discontinuation of anticoagulation in a large cohort of individuals who suffered VTE (EDITH cohort). The primary endpoints were recurrent symptomatic VTE and MACE (composite of non-fatal acute coronary syndrome, stroke and cardiovascular death). Arterial thrombotic event (ATE) exclusively was also considered. Independent predictors of main outcomes were derived from multivariable Cox regression models. Weighted integer points based on the effect estimate of identified predictors were used to derive the final risk scores. A total of 1,999 participants (mean age: 54.78 years, 46.4% male, 43.6% unprovoked VTE) were included in the derivation cohort and 10,000 in the validation cohort (built using bootstrapping). During a median post-anticoagulation follow-up of 6.9 years, recurrent VTE occurred in 29.5% of participants and MACE in 14.8%. Independent predictors of recurrent VTE were male sex, age >65 years, cancer-associated VTE, and unprovoked VTE (vs. transient risk factor-associated VTE). Independent predictors of MACE were age >65 years, cancer-associated VTE, hypertension, renal insufficiency, and atrial fibrillation. The risk of recurrent VTE (moderate vs. low: hazard ratio [HR]: 2.62, 95% confidence interval [CI]: 2.06-3.34; high vs. low: HR: 3.78, 95% CI: 2.91-4.89), MACE (moderate vs. low: HR: 6.37, 95% CI: 3.19-12.69; high vs. low: HR: 12.32, 95% CI: 6.09-24.89), and ATE (based on MACE-predict risk score) increased gradually from the lowest to highest of the respective prediction risk score groups. These results were confirmed in the validation cohort with overall reasonable models' discrimination performance (recurrent VTE C-statistic: 0.62-0.63, MACE and ATE C-statistic: 0.72-0.77). Contemporary simple risk scores based on readily available clinical characteristics can reasonably predict the risk of recurrent VTE and MACE after the discontinuation of anticoagulation. These findings may influence the choice of anticoagulation strategy after the acute phase of VTE and, therefore, need confirmation by further studies.

The advancement of computational tools for cartometric analysis has opened new avenues for the identification and understanding of stemmatic relationships between historical maps through the analysis of their planimetric distortions. The 19th-century Western cartographic depiction of Jerusalem serves as an ideal case study in this context. The challenges of conducting comprehensive onsite surveys—due to limited time and local knowledge—combined with the fascination surrounding the area’s representation, resulted in a proliferation of maps marked by frequent errors, distortions, and extensive copying. How can planimetric similarities and differences between maps be measured, and what insights can be derived from these comparisons? This paper introduces a methodology aimed at detecting and segmenting regions of local planimetric similarity across maps, corresponding to the portions that were either copied between them or derived from a common source. To detect these areas, the ground control points from the georeferencing process are employed to deform a common lattice grid for each map. These grids, triangulated to maintain shape rigidity, can be partitioned under conditions of geometric similarity, allowing for the segmentation and clustering of locally similar regions that represent shared areas between the maps. By integrating this segmentation with a filter on the intensity of distortion, the areas of the grid that are almost non-deformed, and thus not relevant for the study, can be excluded. To showcase the support this methodology offers for close reading, it is applied to the maps in the dataset depicting the Russian Compound. The methodology serves as a tool to assist in constructing the genealogy of the area’s representation and uncovering new historical insights. A larger dataset of 50 maps from the 19th century is then used to identify all the local predecessors of a given map, showcasing another application of the methodology, particularly when working with extensive collections of maps. These findings highlight the potential of computational cartometry to uncover hidden layers of cartographic knowledge and to advance the digital genealogy of map collections.

We construct a quantum Dolbeault double complex ⊕p,qΩp,q on the quantum plane Cq2. This solves the long-standing problem that the standard differential calculus on the quantum plane is not a ∗-calculus, by embedding it as the holomorphic part of a ∗-calculus. We show in general that any Nichols–Woronowicz algebra or braided plane B+(V), where V is an object in an Abelian C-linear braided bar category of real type, is a quantum complex space in this sense of a factorisable Dolbeault double complex. We combine the Chern construction on Ω1,0 in such a Dolbeault complex for an algebra A with its conjugate to construct a canonical metric-compatible connection on Ω1 associated with a class of quantum metrics, and apply this to the quantum plane. We also apply this to finite groups G with Cayley graph generators split into two halves related by inversion, constructing such a Dolbeault complex Ω(G) in this case. This construction recovers the quantum Levi-Civita connection for any edge-symmetric metric on the integer lattice with Ω(Z), now viewed as a quantum complex structure on Z. We also show how to build natural quantum metrics on Ω1,0 and Ω0,1 separately, where the inner product in the case of the quantum plane, in order to descend to ⊗A, is taken with values in an A-bimodule.

We completely characterize point-line configurations with $\Theta(n^{4/3})$ incidences when the point set is a section of the integer lattice. This can be seen as the main special case of the structural Szemerédi-Trotter problem. We also derive a partial characterization for several generalizations: (i) We rule out the concurrent lines case when the point set is a Cartesian product of an arithmetic progression and an arbitrary set. (ii) We study the case of a Cartesian product where one or both sets are generalized arithmetic progression. Our proofs rely on deriving properties of multiplicative energies.