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  1. Abstract

    A dramatic increase in the number of outbreaks of dengue has recently been reported, and climate change is likely to extend the geographical spread of the disease. In this context, this paper shows how a neural network approach can incorporate dengue and COVID-19 data as well as external factors (such as social behaviour or climate variables), to develop predictive models that could improve our knowledge and provide useful tools for health policy makers. Through the use of neural networks with different social and natural parameters, in this paper we define aCorrelation Modelthrough which we show that the number of cases of COVID-19 and dengue have very similar trends. We then illustrate the relevance of our model by extending it to a Long short-term memory model (LSTM) that incorporates both diseases, and using this to estimate dengue infections via COVID-19 data in countries that lack sufficient dengue data.

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  2. Abstract

    This article represents a 1st step toward understanding the well-posedness of the dispersive Hunter–Saxton equation, which arises in the study of nematic liquid crystals. Although the equation has formal similarities with the KdV equation, the lack of $L^2$ control gives it a quasilinear character. Further, the lack of spatial decay obstructs access to dispersive tools, including local smoothing estimates. Here, we give the 1st proof of local and global well-posedness for the Cauchy problem. Secondly, we improve our well-posedness results with respect to the low regularity of the initial data. The key techniques we use include constructing modified energies to realize a normal form analysis in our quasilinear setting, and frequency envelopes to prove continuous dependence with respect to the initial data.

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  3. Abstract

    We consider the Kuramoto–Sivashinsky equation (KSE) on the two-dimensional torus in the presence of advection by a given background shear flow. Under the assumption that the shear has a finite number of critical points and there are linearly growing modes only in the direction of the shear, we prove global existence of solutions with data in$$L^2$$L2, using a bootstrap argument. The initial data can be taken arbitrarily large.

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  4. Abstract We establish the Minimal Model Program for arithmetic threefolds whose residue characteristics are greater than five. In doing this, we generalize the theory of global $F$ F -regularity to mixed characteristic and identify certain stable sections of adjoint line bundles. Finally, by passing to graded rings, we generalize a special case of Fujita’s conjecture to mixed characteristic. 
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    Free, publicly-accessible full text available May 10, 2024
  5. Free, publicly-accessible full text available May 4, 2024
  6. Abstract We consider the discrete defocusing nonlinear Schrödinger equation in its integrable version, which is called defocusing Ablowitz–Ladik lattice. We consider periodic boundary conditions with period N and initial data sampled according to the Generalized Gibbs ensemble. In this setting, the Lax matrix of the Ablowitz–Ladik lattice is a random CMV-periodic matrix and it is related to the Killip-Nenciu Circular $$\beta $$ β -ensemble at high-temperature. We obtain the generalized free energy of the Ablowitz–Ladik lattice and the density of states of the random Lax matrix by establishing a mapping to the one-dimensional log-gas. For the Gibbs measure related to the Hamiltonian of the Ablowitz–Ladik flow, we obtain the density of states via a particular solution of the double-confluent Heun equation. 
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    Free, publicly-accessible full text available May 1, 2024
  7. Free, publicly-accessible full text available May 1, 2024
  8. Abstract. Using totally symmetric sets, Chudnovsky–Kordek–Li–Partin gave a superexponential lower bound on the cardinality of non-abelian finite quotients of the braid group. In this paper, we develop new techniques using multiple totally symmetric sets to count elements in non-abelian finite quotients of the braid group. Using these techniques, we improve the lower bound found by Chudnovsky et al. We exhibit totally symmetric sets in the virtual and welded braid groups and use our new techniques to find superexponential bounds for the finite quotients of the virtual and welded braid groups. 
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    Free, publicly-accessible full text available March 23, 2024
  9. Abstract We investigate maximal tori in the Hochschild cohomology Lie algebra ${\operatorname {HH}}^1(A)$ of a finite dimensional algebra $A$, and their connection with the fundamental groups associated to presentations of $A$. We prove that every maximal torus in ${\operatorname {HH}}^1(A)$ arises as the dual of some fundamental group of $A$, extending the work by Farkas, Green, and Marcos; de la Peña and Saorín; and Le Meur. Combining this with known invariance results for Hochschild cohomology, we deduce that (in rough terms) the largest rank of a fundamental group of $A$ is a derived invariant quantity, and among self-injective algebras, an invariant under stable equivalences of Morita type. Using this we prove that there are only finitely many monomial algebras in any derived equivalence class of finite dimensional algebras; hitherto this was known only for very restricted classes of monomial algebras. 
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  10. Finding roots of univariate polynomials is one of the fundamental tasks of numerics, and there is still a wide gap between root finders that are well understood in theory and those that perform well in practice. We investigate the root-finding method of Weierstrass, also known as the Durand–Kerner-method: this is a root finder that tries to approximate all roots of a given polynomial in parallel. This method has been introduced 130 years ago and has since then a good reputation for finding all roots in practice except in obvious cases of symmetry. Nonetheless, very little is known about its global dynamics and convergence properties. We show that the Weierstrass method, like the well-known Newton method, is not generally convergent: there are open sets of polynomials  p p of every degree d ≥ 3 d \ge 3 such that the dynamics of the Weierstrass method applied to  p p exhibits attracting periodic orbits. Specifically, all polynomials sufficiently close to Z 3 + Z + 175 Z^3 + Z + 175 have attracting cycles of period  4 4 . Here, period 4 4 is minimal: we show that for cubic polynomials, there are no periodic orbits of length 2 2 or  3 3 that attract open sets of starting points. We also establish another convergence problem for the Weierstrass method: for almost every polynomial of degree d ≥ 3 d\ge 3 there are orbits that are defined for all iterates but converge to  ∞ \infty ; this is a problem that does not occur for Newton’s method. Our results are obtained by first interpreting the original problem coming from numerical mathematics in terms of higher-dimensional complex dynamics, then phrasing the question in algebraic terms in such a way that we could finally answer it by applying methods from computer algebra. The main innovation here is the translation into an algebraic question, which is amenable to (exact) computational methods close to the limits of current computer algebra systems. 
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