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Since Kripke, philosophers have distinguished a priori true statements from necessarily true ones. A statement is a priori true if its truth can be established before experience, and necessarily true if it could not have been false according to logical or metaphysical laws. This distinction can be captured formally using two-dimensional semantics.There is a natural way to extend the notions of apriority and necessity so they can also apply to questions. Questions either can or cannot be resolved before experience, and either are or are not about necessary facts. Classical two-dimensionalism has no account of question meanings, so it has to be combined with a framework for question semantics in order to capture these observations. It is shown in [14] how two-dimensional semantics can be combined with inquisitive semantics, in which questions are analyzed in terms of information. The present paper investigates the logic of two-dimensional inquisitive semantics, and provides a complete proof system.

There is widespread excitement in the literature about the method of arbitrary functions: many take it to show that it is from the dynamics of systems that the objectivity of probabilities emerge. In this paper, I differentiate three ways in which a probability function might be objective, and I argue that the method of arbitrary functions cannot help us show that dynamics objectivise probabilities in any of these senses.

We show that a A-linear map of Hilbert A-modules is induced by a unitary Hilbert module operator if and only if it extends to an ordinary unitary on appropriately defined enveloping Hilbert spaces. Applications to the theory of multiplicative unitaries let us to compute the equivalence classes of Hilbert modules over a class of C*-algebraic quantum groups. We, thus, develop a theory that, for example, could be used to show non-existence of certain co-actions. In particular, we show that the Cuntz semigroup functor takes a co-action to a multiplicative action.

The aim of this paper is to study, by using the mathematical tools developed by Chałupnik, Touzé, and Van der Kallen, the effect of the Frobenius twist on $\operatorname{Ext}$-group in the category of strict polynomial functors. As an application, we obtain explicit formulas of cohomology of the orthogonal groups and symplectic ones.

In this paper, we investigate temporal clusters of extremes defined as subsequent exceedances of high thresholds in a stationary time series. Two meaningful features of these clusters are the probability distribution of the cluster size and the ordinal patterns giving the relative positions of the data points within a cluster. Since these patterns take only the ordinal structure of consecutive data points into account, the method is robust under monotone transformations and measurement errors. We verify the existence of the corresponding limit distributions in the framework of regularly varying time series, develop non-parametric estimators and show their asymptotic normality under appropriate mixing conditions. The performance of the estimators is demonstrated in a simulated example and a real data application to discharge data of the river Rhine., Deutsche Forschungsgemeinschaft, Projekt DEAL

Let $\mathcal{M}$ be a $\sigma$-finite von Neumann algebra, equipped with a normal faithful state $\varphi$, and let $\mathcal{A}$ be maximal subdiagonal algebra of $\mathcal{M}$. We prove Stein-Weiss type interpolation theorem of Haagerup noncommutative $H^{p}$-spaces associated with $\A$., Comment: 16. arXiv admin note: text overlap with arXiv:2209.08542

This paper puts forward a new account of rigorous mathematical proof and its epistemology. One novel feature is a focus on how the skill of reading and writing valid proofs is learnt, as a way of understanding what validity itself amounts to. The account is used to address two current questions in the literature: that of how mathematicians are so good at resolving disputes about validity, and that of whether rigorous proofs are necessarily formalizable.

We show that the solution (in the sense of distribution) to the Cauchy problem with the periodic boundary condition associated with the modified Benjamin–Ono equation is unique in $L^\infty _t(H^s(\mathbb{T} ))$ for $s>1/2$. The proof is based on the analysis of a normal form equation obtained by infinitely many reduction steps using integration by parts in time after a suitable gauge transform.

In this paper we develop a general computer-assisted proof method for periodic solutions to delay differential equations. The class of problems considered includes systems of delay differential equations with an arbitrary number of (forward and backward) delays. When the nonlinearities include nonpolynomial terms we introduce auxiliary variables to first rewrite the problem into an equivalent polynomial one. We then apply a flexible fixed point technique in a space of geometrically decaying Fourier coefficients. We showcase the efficacy of this method by proving periodic solutions in the well-known Mackey–Glass delay differential equation for the classical parameter values.

Recently, a symmetric differential operator (SDO) is attracted to studying in the field of mathematical analysis. A new formal of SDO is presented to generalize some well known differential operators in a complex domain. According to this formulation, we shall exam the boundedness and compactness of this operator in complex spaces, such as Hilbert space and Sobolev space. For this purpose, we suggest new norms to solve the fractional Beltrami equation in the open unit disk. This operator has ability to depict the analytic geometric representation of the solution of second order differential equation utilizing the concept of Schwarzian derivative in the open unit disk. Applications in imagings are given the sequel.

A graph database is a digraph whose arcs are labeled with symbols from a fixed alphabet. A regular graph pattern (RGP) is a digraph whose edges are labeled with regular expressions over the alphabet. RGPs model navigational queries for graph databases called conjunctive regular path queries (CRPQs). A match of a CRPQ in the database is witnessed by a special navigational homomorphism of the corresponding RGP to the database. We study the complexity of deciding the existence of a homomorphism between two RGPs. Such homomorphisms model a strong type of containment between the two corresponding CRPQs. We show that this problem can be solved by an EXPTIME algorithm (while general query containmement in this context is EXPSPACE-complete). We also study the problem for restricted RGPs over a unary alphabet, that arise from some applications like XPath or SPARQL. For this case, homomorphism-based CRPQ containment is in NP. We prove that certain interesting cases are in fact polynomial-time solvable., 15 pages. Short version appeared in the proceedings of the 15th Conference on Computability in Europe (CIE 2019)

In this paper, we study approximation properties of single hidden layer neural networks with weights varying on finitely many directions and thresholds from an open interval. We obtain a necessary and at the same time sufficient measure theoretic condition for density of such networks in the space of continuous functions. Further, we prove a density result for neural networks with a specifically constructed activation function and a fixed number of neurons., 12 pages

There are two definitions of Einstein-Finsler spaces introduced by Akbar-Zadeh, which we will show is equal along the integral curves of $I$-invariant projective vector fields. The sub-algebra of the $C$-projective vector fields, leaving the $H$-curvature invariant, has been studied extensively. Here we show on a closed Finsler space with negative definite Ricci curvature reduces to that of Killing vector fields. Moreover, if an Einstein-Finsler space admits such a projective vector field then the flag curvature is constant. Finally, a classification of compact isotropic mean Landsberg manifolds admitting certain projective vector fields is obtained with respect to the sign of Ricci curvature., 25 pages

Background: Sepsis and septic shock are leading causes of mortality world-wide.1 In patients outside the intensive care unit (ICU) a rising qSOFA (quick Sequential Organ Failure Assessment) score correlates with mortality risk.2,3 We sought to investigate if the duration of a qSOFA score ≥ 2 prior to ICU admission further affects outcomes, namely: ICU mortality, in-hospital mortality and length of ICU stay. Method: A retrospective chart review was performed using the electronic ICU database at a quaternary level hospital in Durban, KwaZulu-Natal, examining entries from 1 January 2008 to 31 December 2017. The review included 235 emergency in-hospital adult admissions with suspected infection, of which 144 had a qSOFA score ≥ 2 prior to ICU admission. Results: There was no significant association between the duration of a qSOFA score ≥ 2 prior to ICU admission and ICU mortality (p = 0.975), in-hospital mortality (p = 0.918) and length of ICU stay until demise (p = 0.848) or discharge (p = 0.624). The qSOFA score was significantly associated with ICU mortality with scores of 0, 1, 2 and 3 resulting in ICU mortality rates of 0%, 22.5%, 53.7% and 84.6% respectively (p < 0.001). Conclusion: The duration of a qSOFA score ≥ 2 prior to emergency ICU admission was not significantly associated with ICU mortality, in-hospital mortality or length of ICU stay in adults with suspected infection. The full article is available at https://doi.org/10.36303/SAJAA.2020.26.3.2330

Background: There exist a limited number of studies demonstrating the benefit of initiating anti-retroviral agents (ARVs) for the treatment of human immunodeficiency virus (HIV) in the critical care setting. However, there are physiological, practical, and ethical arguments against making HIV testing and initiation of ARVs routine practice. Methods: A self-administered, cross-sectional survey of healthcare professionals was conducted with data collected using a questionnaire distributed to delegates of the Critical Care Society of South Africa (CCSSA) 2016 Congress, and members of the society. Results: There were 101 respondents. Eight per cent would not consider testing a critically ill patient for HIV (status previously unknown), largely because they considered this unethical. Sixty-nine per cent would consider commencing ARVs in a patient newly-diagnosed with HIV during an inter-current critical illness. The factor most likely to guide them towards this was severity of illness. In general, the greatest concerns focused around biological/medical issues. However, those clinicians who would not consider initiation of ARVs were more likely than those who would, to be concerned about psychosocial issues. Conclusion: Although the majority of clinicians would consider initiation of ARVs acutely in critically ill patients, it is apparent that the clinical decision-making around this matter is complex. Advocacy is needed to further clarify relevant ethical and legal dilemmas, and in the interim consultative and collaborative care is encouraged.

We study nonlinear parabolic stochastic partial differential equations with Wick-power and Wick-polynomial type nonlinearities set in the framework of white noise analysis. These equations include the stochastic Fujita equation, the stochastic Fisher-KPP equation and the stochastic FitzHugh-Nagumo equation among many others. By implementing the theory of $C_0-$semigroups and evolution systems into the chaos expansion theory in infinite dimensional spaces, we prove existence and uniqueness of solutions for this class of SPDEs. In particular, we also treat the linear nonautonomous case and provide several applications featured as stochastic reaction-diffusion equations that arise in biology, medicine and physics.

We show that there is a measure-preserving system $(X,\mathscr{B}, \mu, T)$ together with functions $F_0, F_1, F_2 \in L^{\infty}(\mu)$ such that the correlation sequence $C_{F_0, F_1, F_2}(n) = \int_X F_0 \cdot T^n F_1 \cdot T^{2n} F_2 d\mu$ is not an approximate integral combination of $2$-step nilsequences., Comment: 14 pages

In this paper, we introduce an algorithm for computing p-adic integrals on bad reduction hyperelliptic curves. For bad reduction curves, there are two notions of p-adic integration: Berkovich-Coleman integrals which can be performed locally; and abelian integrals with desirable number-theoretic properties. By covering a bad reduction hyperelliptic curve by annuli and basic wide open sets, we reduce the computation of Berkovich-Coleman integrals to the known algorithms on good reduction hyperelliptic curves. These are due to Balakrishnan, Bradshaw, and Kedlaya, and to Balakrishnan and Besser for regular and meromorphic 1-forms on good reduction curves, respectively. We then employ tropical geometric techniques due to the first-named author with Rabinoff and Zureick-Brown to convert the Berkovich-Coleman integrals into abelian integrals. We provide examples of our algorithm, verifying that certain abelian integrals between torsion points vanish., Comment: Comments Welcome! figure taken from arxiv::1606.09618; v2 minor revisions

Let R be a commutative noetherian ring. The notion of n-wide subcategories of Mod R is introduced and studied in Matsui-Nam-Takahashi-Tri-Yen in relation to the cohomological dimension of a specialization-closed subset of Spec R. In this paper, we introduce the notions of n-coherent subsets of Spec R and n-uniform subcategories of D(Mod R), and explore their interactions with n-wide subcategories of Mod R. We obtain a commutative diagram which yields filtrations of subcategories of Mod R, D(Mod R) and subsets of Spec R and complements classification theorems of subcategories due to Gabriel, Krause, Neeman, Takahashi and Angeleri Hugel-Marks-Stovicek-Takahashi-Vitoria., Comment: 17 pages, to appear in IMRN

Web services are moving toward a new emerging technology lead to the migration of SOAP to RESTful API, which is an Architectural Style that holds Lightweight, Stateless, Uniform Interface, etc., as its constraints. Various sources of clusters of resources, entity, database relations are access throughout the distributed environment across the internet. Generative Power of the RESTful API witnesses the emergence of many companies whose whole business process is based upon the building applications. Since the Syntactic essentials of RESTful Web Services are mainly concerned with the RESTful API, there is a need for evaluation of those essentials whether they are computable or not. The proposed work is carried out on taking resources as simple and effective using Primitive Recursive Function (PRF). Primitive Recursive Function (PRF) uses Turing Machine for REST API capability evaluation and the Service Invocation along with the Application Logic and AppState Logic in order to handle manageability of the RESTful resources via computability evaluation with or without security. To demonstrate the effectiveness of our evaluation process, we conduct a case study on the available REST web services using Primitive Recursive Resources (PRR). The results of our case study show that our evaluation process achieves greater portability, reliability, scalability, etc., which in turn results in high performance.

We study an agglomerative clustering problem motivated by interactive glyphs in geo-visualization. Consider a set of disjoint square glyphs on an interactive map. When the user zooms out, the glyphs grow in size relative to the map, possibly with different speeds. When two glyphs intersect, we wish to replace them by a new glyph that captures the information of the intersecting glyphs. We present a fully dynamic kinetic data structure that maintains a set of $n$ disjoint growing squares. Our data structure uses $O(n (\log n \log\log n)^2)$ space, supports queries in worst case $O(\log^3 n)$ time, and updates in $O(\log^7 n)$ amortized time. This leads to an $O(n\alpha(n)\log^7 n)$ time algorithm to solve the agglomerative clustering problem. This is a significant improvement over the current best $O(n^2)$ time algorithms., Comment: 14 pages, 7 figures

Combining mathematical and physical understanding in reasoning is difficult, and a growing body of research shows that students experience problems with the combination of physics and mathematics in reasoning beyond the introductory level. We investigated students' reasoning about boundary conditions (BCs) for the diffusion equation by conducting exploratory task-based, think-aloud interviews with twelve undergraduate students majoring in physics or mathematics. We identified several difficulties students experienced while solving the interview task and categorized them using the conceptual blending framework. This framework states that in reasoning, people draw from separate input spaces, in this case the mathematics and the physics input space, to form a blended space, where they make connections between elements from these spaces. To identify difficulties, we used open coding techniques. We observed few difficulties in the physics space. In the mathematics space, we identified several difficulties that we clustered in two main groups: findings about the mathematical meaning of BCs, and findings about reasoning with functions of two variables. Lastly, we identified four ways in which blending failed. Starting from our findings, we formulate recommendations for teaching and future research.

Using the approach of so-called c-sequences introduced by the fifth author in his recent work, we give much simpler and shorter proofs of multivalued Perov's type results with respect to the ones presented in the recently published paper by M. Abbas et al. Our proofs improve, complement, unify and enrich the ones from the recent papers. Further, in the last section of this paper, we correct and generalize the well-known Perov's fixed point result. We show that this result is in fact equivalent to Banach's contraction principle.

The directed power graph $$\vec {\mathcal {G}}( G)$$ of a group G is the simple digraph with vertex set G in which $$x\rightarrow y$$ if y is a power of x, and the power graph is the underlying simple graph $$\mathcal {G}( G)$$ . In this paper, three versions of the definition of the power graph are discussed, and it is proved that the power graph by any of the three versions of the definition determines the other two up to isomorphism. It is also proved that if G is a torsion-free group of nilpotency class 2 and if H is a group such that $$\mathcal {G}( H)\cong \mathcal {G}( G)$$ , then G and H have isomorphic directed power graphs, which was an open problem proposed by Cameron, Guerra and Jurina [9].

In this paper, we extend the notion of microstate free entropy to the bi-free setting. In particular, using the bi-free analogue of random matrices, microstate bi-free entropy is defined. Properties essential to an entropy theory are developed, such as the behaviour of the entropy when transformations on the left variables or on the right variables are performed. In addition, the microstate bi-free entropy is demonstrated to be additive over bi-free collections provided additional regularity assumptions are included and is computed for all bi-free central limit distributions. Moreover, an orbital version of bi-free entropy is examined which provides a tighter upper bound for the subadditivity of microstate bi-free entropy and provides an alternate characterization of bi-freeness in certain settings., Comment: (new version contains corrections, orbital bi-free entropy, and a new characterization of bi-freenness)

We assume that the diffusion $X$ satisfies a stochastic differential equation of the form: $dX_t=\mu(X_t, \theta)dt+\sigma_0\nu(X_t)dW_t$, with unknown drift parameter $\theta$ and known diffusion coefficient parameter $\sigma_0$. We prove that approximate maximum likelihood estimator of drift parameter $\bar{; ; ; ; ; ; \theta}; ; ; ; ; ; _n$ obtained from discrete observations $(X_{; ; ; ; ; ; i\Delta_n}; ; ; ; ; ; , 0\leq i\leq n)$ along fixed time interval $[0, T]$, and when $\Delta_n =\frac{; ; ; ; ; ; T}; ; ; ; ; ; {; ; ; ; ; ; n}; ; ; ; ; ; $ tends to zero, is locally asymptotic mixed normal, with covariance matrix which depends on MLE $\hat{; ; ; ; ; ; \theta}; ; ; ; ; ; $ obtained from continuous observations $(X_t, 0\leq t\leq T)$ along fixed time interval $[0, T]$, and on path $(X_t, 0\leq t\leq T)$.

We prove that a solution of an elliptic operator with periodic coefficients behaves on large scales like an analytic function, in the sense of approximation by polynomials with periodic corrections. Equivalently, the constants in the large-scale $C^{k,1}$ estimate scale exponentially in $k$, just as for the classical estimate for harmonic functions. As a consequence, we characterize entire solutions of periodic, uniformly elliptic equations which exhibit growth like $O(\exp(\delta|x|))$ for small~$\delta>0$. The large-scale analyticity also implies quantitative unique continuation results, namely a three-ball theorem with an optimal error term as well as a proof of the nonexistence of $L^2$ eigenfunctions at the bottom of the spectrum., Comment: 34 pages

The Kuramoto–Sinelshchikov–Cahn–Hilliard equation models the spinodal decomposition of phase separating systems in an external field, the spatiotemporal evolution of the morphology of steps on crystal surfaces and the growth of thermodynamically unstable crystal surfaces with strongly anisotropic surface tension. In this paper, we prove the well-posedness of the Cauchy problem, associated with this equation.

We introduce a distributionally robust minimium mean square error estimation model with a Wasserstein ambiguity set to recover an unknown signal from a noisy observation. The proposed model can be viewed as a zero-sum game between a statistician choosing an estimator—that is, a measurable function of the observation—and a fictitious adversary choosing a prior—that is, a pair of signal and noise distributions ranging over independent Wasserstein balls—with the goal to minimize and maximize the expected squared estimation error, respectively. We show that, if the Wasserstein balls are centered at normal distributions, then the zero-sum game admits a Nash equilibrium, by which the players’ optimal strategies are given by an affine estimator and a normal prior, respectively. We further prove that this Nash equilibrium can be computed by solving a tractable convex program. Finally, we develop a Frank–Wolfe algorithm that can solve this convex program orders of magnitude faster than state-of-the-art general-purpose solvers. We show that this algorithm enjoys a linear convergence rate and that its direction-finding subproblems can be solved in quasi-closed form. Funding: This research was supported by the Swiss National Science Foundation [Grants BSCGI0_ 157733 and 51NF40_180545], an Early Postdoc.Mobility Fellowship [Grant P2ELP2_195149], and the European Research Council [Grant TRUST-949796].

Using umbral calculus, we establish a symmetric identity for any sequence of polynomials satisfying A′ n +1(x) = (n + 1)An (x) with A 0(x) a constant polynomial. This identity allows us to obtain in a simple way some known relations involving Apostol-Bernoulli polynomials, Apostol--Euler polynomials and generalized Bernoulli polynomials attached to a primitive Dirichlet character.

The aim of this article is to investigate two new classes of quaternions, namely, balancing and Lucas-balancing quaternions that are based on balancing and Lucas-balancing numbers, respectively. Further, some identities including Binet’s formulas, summation formulas, Catalan’s identity, etc. concerning these quaternions are also established.

We complete the discrete cluster categories of type A as defined by Igusa and Todorov, by embedding such a discrete cluster category inside a larger one, and then taking a certain Verdier quotient. The resulting category is a Hom‐finite Krull–Schmidt triangulated category containing the discrete cluster category as a full subcategory. The objects and Hom‐spaces in this new category can be described geometrically, even though the category is not 2‐Calabi–Yau and Ext‐spaces are not always symmetric. We describe all cluster‐tilting subcategories. Given such a subcategory, we define a cluster character that takes values in a ring with infinitely many indeterminates. Our cluster character is new in that it takes into account infinite‐dimensional subrepresentations of infinite‐dimensional ones. We show that it satisfies the multiplication formula and also the exchange formula, provided that the objects being exchanged satisfy some local Calabi–Yau conditions.

We continue the investigation of coupled Sachdev-Ye-Kitaev(SYK) models without Schwarzian action dominance. Like the original SYK, at large N and low energies these models have an approximate reparametrization symmetry. However, the dominant action for reparametrizations is non-local due to the presence of irrelevant local operator with small conformal dimension. We semi-analytically study different thermodynamic properties and the 4-point function and demonstrate that they significantly differ from the Schwarzian prediction. However, the residual entropy and maximal chaos exponent are the same as in Majorana SYK. We also discuss chain models and finite N corrections., v1: 48 pages, 24 figures. v2: typos corrected, references added

Coalition announcement logic (CAL) is one of the family of the logics of quantified announcements. It allows us to reason about what a coalition of agents can achieve by making announcements in the setting where the anti-coalition may have an announcement of their own to preclude the former from reaching its epistemic goals. In this paper, we describe a PSPACE-complete model checking algorithm for CAL that produces winning strategies for coalitions. The algorithm is implemented in a proof-of-concept model checker.

We prove an algebraicity of special values of symmetric 4-th and 6-th power L-functions of Hilbert cusp forms using its symmetric power lifting and several algebraicity results on special values of automorphic L-functions.

We take an elementary and systematic approach to the problem of extending the Tutte polynomial to the setting of embedded graphs. Four notions of embedded graphs arise naturally when considering deletion and contraction operations on graphs on surfaces. We give a description of each class in terms of coloured ribbon graphs. We then identify a universal deletion-contraction invariant (i.e., a `Tutte polynomial') for each class. We relate these to graph polynomials in the literature, including the Bollob\'as--Riordan, Krushkal, and Las Vergnas polynomials, and give state-sum formulations, duality relations, deleton-contraction relations, and quasi-tree expansions for each of them., Comment: This is the authors accepted manuscript / version of record with an unfortunate typo corrected on p24

This paper focuses on the spherical fixed point set of a circle action on an ALE space endowed with Kronheimer's hyperkaehler metric. The induced metric on the sphere is described by using the algebraic geometry of rational curves on algebraic surfaces, in particular the lines on a cubic., Dedicated to the memory of Michael Atiyah

International audience; In the first edition of this book, we discussed methods for determining the support of an inhomogeneous media from measured far field data as well as an extensive study of the central role played by the transmission eigenvalue problem in the mathematical development of these methods. In particular, we introduced the generalized linear sampling method (GLSM) and showed that this method provides a mathematical explanation of why it is permissible to use Tikhonov regularization to obtain an approximate solution of the far field equation associated with the linear sampling method.In the five years since the first edition of our book appeared, there has been considerable progress in both the development of GLSM as well as the theory of transmission eigenvalues. In this second edition, in addition to correcting typos in the first edition, we have added several highlights taken from these new developments. In particular, we have included new chapters on 1) the use of modified background media in the nondestructive testing of materials and in particular methods for determining the modified transmission eigenvalues that arise in such applications from measured far field data, 2) a study of a subset of transmission eigenvalues, called non-scattering wave numbers, through the use of techniques taken from the theory of free boundary value problems for elliptic partial differential equations and 3) the duality between scattering poles and transmission eigenvalues which, in addition to their intrinsic mathematical interest, leads to new methods for the numerical computation of scattering poles.

V. V. Grushin and V. P. Palamodov proved in 1962 that it is impossible to place in [Formula: see text] uncountably many pairwise disjoint polyhedra each homeomorphic to the Moebius band. We generalize this result in two directions. First, we give a generalization of this result to tame subsets in [Formula: see text], [Formula: see text]. Second, we show that in case of [Formula: see text] the theorem holds for arbitrarily topologically embedded (not necessarily tame) Moebius bands.

Let $(\mathfrak{g},\tau)$ be a real simple symmetric Lie algebra and let $W \subset \mathfrak{g}$ be an invariant closed convex cone which is pointed and generating with $\tau(W) = -W$. For elements $h \in \mathfrak{g}$ with $\tau(h) = h$, we classify the Lie algebras $\mathfrak{g}(W,\tau,h)$ which are generated by the closed convex cones \[C_{\pm}(W,\tau,h) := (\pm W) \cap \mathfrak{g}_{\pm 1}^{-\tau}(h),\] where $\mathfrak{g}^{-\tau}_{\pm 1}(h) := \{x \in \mathfrak{g} : \tau(x) = -x, [h,x] = \pm x\}$. These cones occur naturally as the skew-symmetric parts of the Lie wedges of endomorphism semigroups of certain standard subspaces. We prove in particular that, if $\mathfrak{g}(W,\tau,h)$ is non-trivial, then it is either a hermitian simple Lie algebra of tube type or a direct sum of two Lie algebras of this type. Moreover, we give for each hermitian simple Lie algebra and each equivalence class of involutive automorphisms $\tau$ of $\mathfrak{g}$ with $\tau(W) = -W$ a list of possible subalgebras $\mathfrak{g}(W,\tau,h)$ up to isomorphy., Comment: The title of the paper has been changed; the introduction has been rewritten; some of the proofs have been shortened

Let $\mathrm{k}_{0}$ be a field and $W$ a finite orthogonal reflection group over $\mathrm{k}_{0}$. We prove Serre's splitting principle for cohomological invariants of $W$ with values in Rost's cycle modules (over $\mathrm{k}_{0}$) if the characteristic of $\mathrm{k}_{0}$ is coprime to $|W|$. We then show that this principle for such groups holds also for Witt- and Milnor-Witt $K$-theory invariants., 20 pages

We investigate two sparse Sachdev-Ye-Kitaev (SYK) systems coupled by a bilinear term as a holographic quantum mechanical description of an eternal traversable wormhole in the low temperature limit. Each SYK system consists of $N$ Majorana fermions coupled by random $q$-body interactions. The degree of sparseness is captured by a regular hypergraph in such a way that the Hamiltonian contains exactly $k\,N$ independent terms. We improve on the theoretical understanding of the sparseness property by using known measures of hypergraph expansion. We show that the sparse version of the two coupled SYK model is gapped with a ground state close to a thermofield double state. Using Krylov subspace and parallelization techniques, we simulate the system for $q=4$ and $q=8.$ The sparsity of the model allows us to explore larger values of $N$ than the ones existing in the literature for the all-to-all SYK. We analyze in detail the two-point functions and the transmission amplitude of signals between the two systems. We identify a range of parameters where revivals obey the scaling predicted by holography and signals can be interpreted as traversing the wormhole., Comment: 28 pages, 11 figures, published version

We study the Stokes problem for the incompressible fluid with mixed nonlinear boundary conditions of subdifferential type. The latter involve a unilateral boundary condition, the Navier slip condition, a nonmonotone version of the nonlinear Navier–Fujita slip condition, and the threshold slip and leak condition of frictional type. The weak form of the problem leads to a new class of variational–hemivariational inequalities on convex sets for the velocity field. Solution existence and the weak compactness of the solution set to the inequality problem are established based on the Schauder fixed point theorem.

We give a self-contained survey of some approaches aimed at a global description of the geometry underlying double field theory. After reviewing the geometry of Courant algebroids and their incarnations in the AKSZ construction, we develop the theory of metric algebroids including their graded geometry. We use metric algebroids to give a global description of doubled geometry, incorporating the section constraint, as well as an AKSZ-type construction of topological doubled sigma-models. When these notions are combined with ingredients of para-Hermitian geometry, we demonstrate how they reproduce kinematical features of double field theory from a global perspective, including solutions of the section constraint for Riemannian foliated doubled manifolds, as well as a natural notion of generalized T-duality for polarized doubled manifolds. We describe the $L_\infty$-algebras of symmetries of a doubled geometry, and briefly discuss other proposals for global doubled geometry in the literature., Comment: 56 pages; v2: minor corrections and additions; v3: minor corrections, presentation improved, reference added; Final version to appear in the Special Issue of Complex Manifolds on "Generalized Geometry"

International audience; Cardiac memory, also known as the Chatterjee phenomenon, refers to the persistent but reversible T-wave changes on ECG caused by an abnormal electrical activation pattern. After a period of abnormal ventricular activation in which the myocardial repolarization is altered and delayed (such as with artificial pacemakers, tachyarrhythmias with wide QRS complexes or ventricular pre-excitation), the heart remembers and mirrors its repolarization in the direction of the vector of "abnormally" activated QRS complexes. This phenomenon alters patterns of gap junction distribution and generates changes in repolarization seen at the level of ionic-channel, ionic concentrations, ionic-current gating and action potential. In this work, we propose a mathematical model of cardiac electrophysiology which takes into account cardiac memory phenomena. The electrical activity in heart through torso, which is dependent on the prior history of accrued heartbeats, is mathematically modeled by a modified bidomain system with time fractional-order dynamics (which are used to describe processes that exhibit memory). This new bidomain system, that I name "it memory bidomain system", is a degenerate nonlinear coupled system of reaction-diffusion equations in shape of a fractional-order differential equation coupled with a set of time fractional-order partial differential equations. Cardiac memory is represented via fractional-order capacitor (associate to capacitive current) and fractional-order cellular membrane dynamics. First, mathematical model is introduced. Afterward, results on generalized Gronwall inequality within the framework of coupled fractional differential equations are developed. Next, the existence and uniqueness of solution of state system are proved as well as stability result. Further, some preliminary numerical applications are performed to show that memory reproduced by fractional-order derivatives can play a significant role on key dependent electrical properties including APD, action potential morphology and spontaneous activity.

Purpose: To compare onset times of glaucoma progression among different glaucoma tests: disc photography (DP), visual field (VF) testing, 2-dimensional (2D) retinal nerve fiber layer (RNFL) thickness, and 3-dimensional (3D) spectral-domain (SD) OCT neuroretinal rim measurements. Design: Prospective, longitudinal cohort study. Participants: One hundred twenty-four eyes of 124 patients with open-angle glaucoma. Methods: Over a 5-year period, 124 patients with open-angle glaucoma underwent yearly DP, VF testing, SD OCT RNFL thickness scans, and optic nerve volume scans (Spectralis; Heidelberg Engineering), all performed on the same day. From high-density optic nerve volume scans, custom-built software calculated the minimum distance band (MDB) thickness, a 3D neuroretinal rim parameter. Patients were classified as glaucoma progressors or nonglaucoma progressors using event-based analysis. Progression by DP and VF testing occurred when 3 masked glaucoma specialists unanimously concurred. Progression by RNFL and MDB thickness occurred if change of more than test–retest variability was observed. Kaplan-Meier curves were constructed to analyze time-to-progression data. Kappa Coefficients were used to measure agreement of progressing eyes among methods. Main Outcome Measures: Time to glaucoma progression among all 4 methods. Results: Global MDB thickness detected glaucoma progression in the highest percentage of eyes (52.4%) compared with DP (16.1%; P < 0.001) and global RNFL thickness (15.3%; P < 0.001). Global MDB thickness detected glaucoma progression earlier than either DP (23 months vs. 44 months; P < 0.001) or global RNFL thickness (23 months vs. 33 months; P < 0.001). Among MDB progressing eyes, 46.2% were confirmed simultaneously or later by other conventional methods. Agreement of glaucoma-progressing eyes for all 4 methods in paired fashion were slight to fair (κ = 0.095–0.300). Conclusions: High-density 3D SD OCT neuroretinal rim measurements detected glaucoma progression approximately 1 to 2 years earlier compared with current clinically available structural tests (i.e., DP and 2D RNFL thickness measurements).

This paper presents a simple decidable logic of functional dependence LFD, based on an extension of classical propositional logic with dependence atoms plus dependence quantifiers treated as modalities, within the setting of generalized assignment semantics for first order logic. The expressive strength, complete proof calculus and meta-properties of LFD are explored. Various language extensions are presented as well, up to undecidable modal-style logics for independence and dynamic logics of changing dependence models. Finally, more concrete settings for dependence are discussed: continuous dependence in topological models, linear dependence in vector spaces, and temporal dependence in dynamical systems and games., 56 pages. Journal of Philosophical Logic (2021)

A complex K3 surface or an algebraic K3 surface in characteristics distinct from $2$ cannot have more than $16$ disjoint nodal curves., A complex K3 surface or an algebraic K3 surface in characteristics distinct from 2 cannot have more than 16 disjoint nodal curves.

In this paper we study a class of impulsive systems of stochastic differential equations with infinite Brownian motions. Sufficient conditions for the existence and uniqueness of solutions are established by mean of some fixed point theorems in vector Banach spaces. An example is provided to illustrate the theory.

The homotopy types of gauge groups of principal $$\mathrm{SO}(4)$$ SO ( 4 ) -bundles over $$S^{4}$$ S 4 are classified p-locally for every prime p, and partial results are obtained integrally. The method generalizes to deal with any quotient of the form $$(S^{3})^{n}/Z$$ ( S 3 ) n / Z where Z is a subgroup generated by $$(-1,\ldots ,-1)$$ ( - 1 , … , - 1 ) .