Max Planck Institute for Mathematics in the Sciences

Graphs have become a commonly used model to study technological, biological, and social systems. Various methods have been proposed to measure graphs’ structural and dynamical properties, providing insights into the fundamental processes and interactions that govern the behavior of these systems. Matrix functions are powerful mathematical tools for assessing vertex centrality, communicability, and diffusion processes. Let $\textbf{M}$ be the adjacency matrix of a weighted undirected graph. Then, the trace of matrix functions, $\boldsymbol{{{\,\textrm{tr}\,}}}(\boldsymbol{f}(\textbf{M}))$, provides insights into global network structural and dynamical properties. Although $\boldsymbol{{{\,\textrm{tr}\,}}}(\boldsymbol{f}(\textbf{M}))$ can be computed using the diagonalization method for graphs with a few thousand vertices, this approach is impractical for large-scale networks due to its computational complexity. Here, we present a message-passing method to approximate $\boldsymbol{{{\,\textrm{tr}\,}}}(\boldsymbol{f}(\textbf{M}))$ for graphs with short cycles that runs in linear time up to logarithmic terms. We compare our proposal with the state-of-the-art approach through simulations and real-world network applications, achieving comparable accuracy in less time.

Quantum steering, an intermediate quantum correlation lying between entanglement and nonlocality, has emerged as a critical quantum resource for a variety of quantum information processing tasks such as quantum key distribution and true randomness generation. The ability to detect and quantify quantum steering is crucial for these applications. Semi‐definite programming (SDP) has proven to be a valuable tool to quantify quantum steering. However, the challenge lies in the fact that the optimal measurement strategy is not priori known, making it time‐consuming to compute the steerable measure for any given quantum state. Furthermore, the utilization of SDP requires full information of the quantum state, necessitating quantum state tomography, which can be complex and resource‐consuming. In this work, the semi‐supervised self‐training model is used to estimate the steerable weight, a pivotal measure of quantum steering. The model can be trained using a limited amount of labeled data, thus reducing the time for labeling. The features are constructed by the probabilities derived by performing three sets of projective measurements under arbitrary local unitary transformations on the target states, circumventing the need for quantum tomography. The model demonstrates robust generalization capabilities and can achieve high levels of precision with limited resources.

Given a residually connected incidence geometry that satisfies two conditions, denoted and , we construct a new geometry with properties similar to those of . This new geometry is inspired by a construction of Lefèvre-Percsy, Percsy and Leemans (Bull Belg Math Soc Simon Stevin 7(4):583–610, 2000). We show how relates to the classical halving operation on polytopes, allowing us to generalize the halving operation to a broader class of geometries, that we call non-degenerate leaf hypertopes. Finally, we apply this generalization to cubic toroids in order to generate new examples of regular hypertopes.

We introduce the notion of $\Theta$-positivity in real semisimple Lie groups. This notion at the same time generalizes Lusztig’s total positivity in split real Lie groups and invariant orders in Lie groups of Hermitian type. We show that there are four families of simple Lie groups which admit a positive structure relative to a subset $\Theta$ of simple roots, and investigate fundamental properties of $\Theta$-positivity. We define and describe the positive and nonnegative unipotent semigroups and show that they give rise to a notion of positive n-tuples in flag varieties.

We prove geometric upper bounds for the Poincaré and Logarithmic Sobolev constants for Brownian motion on manifolds with sticky reflecting boundary diffusion i.e. extended Wentzell-type boundary condition under general curvature assumptions on the manifold and its boundary. The method is based on an interpolation involving energy interactions between the boundary and the interior of the manifold. As side results we obtain explicit geometric bounds on the first nontrivial Steklov eigenvalue, for the norm of the boundary trace operator on Sobolev functions, and on the boundary trace logarithmic Sobolev constant. The case of Brownian motion with pure sticky reflection is also treated.

Multipartite entanglement is a crucial resource for a wide range of quantum information processing tasks, including quantum metrology, quantum computing, and quantum communication. The verification of multipartite entanglement, along with an understanding of its intrinsic structure, is of fundamental importance, both for the foundations of quantum mechanics and for the practical applications of quantum information technologies. Nonetheless, detecting entanglement structures remains a significant challenge, particularly for general states and large‐scale quantum systems. To address this issue, an efficient algorithm that combines semidefinite programming with a gradient descent method is developed. This algorithm is designed to explore the entanglement structure by examining the inner polytope of the convex set that encompasses all states sharing the same entanglement properties. Through detailed examples, it is demonstrated that the superior performance of this approach compared to many of the best‐known methods available today. This method not only improves entanglement detection but also provides deeper insights into the complex structures of many‐body quantum systems, which is essential for advancing quantum technologies.

Big data technology has profoundly transformed the design, development, and operational models of the game industry, ushering in both opportunities and challenges. With the Chinese game market now surpassing 310 billion yuan in size and continuing to grow, the influence of big data is more significant than ever. This paper aims to delve into the challenges that gaming companies face in this data-driven era, using Tencent Games as a case study. The research focuses on three critical issues: First, the problem of game lag is caused by the massive volume of data processing in game development and operation. As games become more complex and data-driven, ensuring seamless performance becomes increasingly difficult, leading to frustrating experiences for players. Second, the study addresses digital ethics concerns, particularly the use of big data in ways that might infringe on user privacy or promote unhealthy gaming habits. Lastly, the paper examines the risks associated with an over-reliance on big data for decision-making, where companies may prioritize data-driven insights at the expense of creativity and innovation. By analyzing the root causes of these challenges, this paper provides practical and effective solutions that can help gaming companies like Tencent navigate the complexities of the big data era while maintaining a competitive edge and ethical standards.

The Schmidt number characterizes the quantum entanglement of a bipartite mixed state and plays a significant role in certifying entanglement of quantum states. We derive a Schmidt number criterion based on the trace norm of the correlation matrix obtained from the general symmetric informationally complete measurements. The criterion gives an effective way to quantify the entanglement dimension of a bipartite state with arbitrary local dimensions. We show that this Schmidt number criterion is more effective and superior than other criteria such as fidelity, CCNR (computable cross-norm or realignment), MUB (mutually unbiased bases) and EAM (equiangular measurements) criteria in certifying the Schmidt numbers by detailed examples.

Many publicly available databases provide disease related data, that makes it possible to link genomic data to medical and meta-data. The cancer genome atlas (TCGA), for example, compiles tens of thousand of datasets covering a wide array of cancer types. Here we introduce an interactive and highly automatized TCGA-based workflow that links and analyses epigenomic and transcriptomic data with treatment and survival data in order to identify possible biomarkers that indicate treatment success. TREMSUCS-TCGA is flexible with respect to type of cancer and treatment and provides standard methods for differential expression analysis or DMR detection. Furthermore, it makes it possible to examine several cancer types together in a pan-cancer type approach. Parallelisation and reproducibility of all steps is ensured with the workflowmanagement system Snakemake. TREMSUCS-TCGA produces a comprehensive single report file which holds all relevant results in descriptive and tabular form that can be explored in an interactive manner. As a showcase application we describe a comprehensive analysis of the available data for the combination of patients with squamous cell carcinomas of head and neck, cervix and lung treated with cisplatin, carboplatin and the combination of carboplatin and paclitaxel. The best ranked biomarker candidates are discussed in the light of the existing literature, indicating plausible causal relationships to the relevant cancer entities.

Quantum entanglement plays a pivotal role in quantum information processing. Quantifying quantum entanglement is a challenging and essential research area within the field. This manuscript explores the relationships between bipartite entanglement concurrence, multipartite entanglement concurrence, and genuine multipartite entanglement (GME) concurrence. We derive lower bounds on GME concurrence from these relationships, demonstrating their superiority over existing results through rigorous proofs and numerical examples. Additionally, we investigate the connections between GME concurrence and other entanglement measures, such as tangle and global negativity, in multipartite quantum systems.

We show higher integrability of minimisers of functionals $\begin{aligned} I(u) = \int _{\Omega } f(x,u(x)) ~d x \end{aligned}$ I ( u ) = ∫ Ω f ( x , u ( x ) ) d x subject to a differential constraint ${\mathscr {A}} u=0$ A u = 0 under natural p -growth and p -coercivity conditions for f and regularity assumptions on $\Omega$ Ω . For the differential operator ${\mathscr {A}}$ A we asssume a rather abstract truncation property that, for instance, holds for operators ${\mathscr {A}}=\textrm{curl}$ A = curl and ${\mathscr {A}}=\textrm{div}$ A = div . The proofs are based on the comparison of the minimiser to the truncated version of the minimiser.

Given a regular covering map φ : Λ → Γ \varphi\colon\Lambda\to\Gamma of graphs, we investigate the subgroup LAut ⁡ ( φ ) \operatorname{LAut}(\varphi) of the automorphism group Aut ⁡ ( A Γ ) \operatorname{Aut}(A_{\Gamma}) of the right-angled Artin group A Γ A_{\Gamma} . This subgroup comprises all automorphisms that can be lifted to automorphisms of A Λ A_{\Lambda} . We first show that LAut ⁡ ( φ ) \operatorname{LAut}(\varphi) is generated by a finite subset of Laurence’s elementary automorphisms. For the subgroup FAut ⁡ ( φ ) \operatorname{FAut}(\varphi) of Aut ⁡ ( A Λ ) \operatorname{Aut}(A_{\Lambda}) that consists of lifts of automorphisms in LAut ⁡ ( φ ) \operatorname{LAut}(\varphi) , there exists a natural homomorphism FAut ⁡ ( φ ) → LAut ⁡ ( φ ) \operatorname{FAut}(\varphi)\to\operatorname{LAut}(\varphi) induced by 𝜑. We then show that the kernel of this homomorphism is virtually a subgroup of the Torelli subgroup IA ⁡ ( A Λ ) \operatorname{IA}(A_{\Lambda}) and deduce a short exact sequence reminiscent of results from the Birman–Hilden theory for surfaces.

The difference set of an outcome in an auction is the set of types that the auction mechanism maps to the outcome. We give a complete characterization of the geometry of the difference sets that can appear for a dominant strategy incentive compatible multi-unit auction showing that they correspond to regular subdivisions of the unit cube. Similarly, we describe the geometry for affine maximizers for n players and m items, showing that they correspond to regular subdivisions of the m -fold product of $(n-1)$ ( n - 1 ) -dimensional simplices. These observations are then used to construct mechanisms that are robust in the sense that the sets of items allocated to the players change only slightly when the players’ reported types are changed slightly.

In this paper, we derive global bounds for the Hölder norms of the gradient of solutions of graphic mean curvature flows with boundaries of arbitrary codimension.

We study the notion of degeneration for affine schemes associated with systems of algebraic differential equations with coefficients in the fraction field of a multivariate formal power series ring. To do this, we use an integral structure of this field that arises as the unit ball associated with the tropical valuation, first introduced in the context of tropical differential algebra. This unit ball turns out to be a particular type of integral domain, known as Bézout domain. By applying to these systems a translation map along a vector of weights that emulates the one used in classical tropical algebraic geometry, the resulting translated systems will have coefficients in this unit ball. When the resulting quotient module over the unit ball is torsion free, then it gives rise to integral models of the original system in which every prime ideal of the unit ball defines an initial degeneration, and they can be found as a base change to the residue field of the prime ideal. In particular, the closed fibres of our integral models can be rightfully called initial degenerations, since we show that there is a bijection between maximal ideals of this unit ball and monomial orders. We use this correspondence to define initial forms of differential polynomials and initial ideals of differential ideals, and we show that they share many features of their classical analogues.

S. Gukov and C. Vafa proposed a characterization of rational superconformal field theories (SCFTs) in dimensions with Ricci‐flat Kähler target spaces in terms of the Hodge structure of the target space, extending an earlier observation by G. Moore. The idea is refined, and a conjectural statement on necessary and sufficient conditions for such SCFTs to be rational is obtained, which is indeed proven to be true in the case the target space is . In the refined statement, the algebraicity of the geometric data of the target space turns out to be essential, and the Strominger–Yau–Zaslow fibration in the mirror correspondence also plays a vital role.