Different types of integrability and their relation to decoherence in central spin models

We investigate the relation between integrability and decoherence in central spin models with more than one central spin. We show that there is a transition between integrability ensured by the Bethe ansatz and integrability ensured by complete sets of commuting operators. This has a significant impact on the decoherence properties of the system, suggesting that it is not necessarily integrability or nonintegrability which is related to decoherence, but rather its type or a change from integrability to nonintegrability.Comment: 4 pages, 3 figure

Integer Polynomial Optimization in Fixed Dimension

We classify, according to their computational complexity, integer optimization problems whose constraints and objective functions are polynomials with integer coefficients and the number of variables is fixed. For the optimization of an integer polynomial over the lattice points of a convex polytope, we show an algorithm to compute lower and upper bounds for the optimal value. For polynomials that are non-negative over the polytope, these sequences of bounds lead to a fully polynomial-time approximation scheme for the optimization problem.Comment: In this revised version we include a stronger complexity bound on our algorithm. Our algorithm is in fact an FPTAS (fully polynomial-time approximation scheme) to maximize a non-negative integer polynomial over the lattice points of a polytop

Parametric Polyhedra with at least kk Lattice Points: Their Semigroup Structure and the k-Frobenius Problem

Given an integral $d \times n$ matrix $A$, the well-studied affine semigroup \mbox{ Sg} (A)=\{ b : Ax=b, \ x \in {\mathbb Z}^n, x \geq 0\} can be stratified by the number of lattice points inside the parametric polyhedra $P_A(b)=\{x: Ax=b, x\geq0\}$. Such families of parametric polyhedra appear in many areas of combinatorics, convex geometry, algebra and number theory. The key themes of this paper are: (1) A structure theory that characterizes precisely the subset \mbox{ Sg}_{\geq k}(A) of all vectors b \in \mbox{ Sg}(A) such that $P_A(b) \cap {\mathbb Z}^n$ has at least $k$ solutions. We demonstrate that this set is finitely generated, it is a union of translated copies of a semigroup which can be computed explicitly via Hilbert bases computations. Related results can be derived for those right-hand-side vectors $b$ for which $P_A(b) \cap {\mathbb Z}^n$ has exactly $k$ solutions or fewer than $k$ solutions. (2) A computational complexity theory. We show that, when $n$, $k$ are fixed natural numbers, one can compute in polynomial time an encoding of \mbox{ Sg}_{\geq k}(A) as a multivariate generating function, using a short sum of rational functions. As a consequence, one can identify all right-hand-side vectors of bounded norm that have at least $k$ solutions. (3) Applications and computation for the $k$-Frobenius numbers. Using Generating functions we prove that for fixed $n,k$ the $k$-Frobenius number can be computed in polynomial time. This generalizes a well-known result for $k=1$ by R. Kannan. Using some adaptation of dynamic programming we show some practical computations of $k$-Frobenius numbers and their relatives

Nonlinear Integer Programming

Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic. The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Numerous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms. We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems. It is our belief that the study of algorithms motivated by theoretical considerations and those motivated by our desire to solve practical instances should and do inform one another. So it is with this viewpoint that we present the subject, and it is in this direction that we hope to spark further research.Comment: 57 pages. To appear in: M. J\"unger, T. Liebling, D. Naddef, G. Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey (eds.), 50 Years of Integer Programming 1958--2008: The Early Years and State-of-the-Art Surveys, Springer-Verlag, 2009, ISBN 354068274

Pruning Algorithms for Pretropisms of Newton Polytopes

Pretropisms are candidates for the leading exponents of Puiseux series that represent solutions of polynomial systems. To find pretropisms, we propose an exact gift wrapping algorithm to prune the tree of edges of a tuple of Newton polytopes. We prefer exact arithmetic not only because of the exact input and the degrees of the output, but because of the often unpredictable growth of the coordinates in the face normals, even for polytopes in generic position. We provide experimental results with our preliminary implementation in Sage that compare favorably with the pruning method that relies only on cone intersections.Comment: exact, gift wrapping, Newton polytope, pretropism, tree pruning, accepted for presentation at Computer Algebra in Scientific Computing, CASC 201

Hypercholesterolemia and 27-Hydroxycholesterol Increase S100A8 and RAGE Expression in the Brain: a Link Between Cholesterol, Alarmins, and Neurodegeneration

Alterations in cholesterol metabolism in the brain have a major role in the physiology of Alzheimer’s disease (AD). Oxysterols are cholesterol metabolites with multiple implications in memory functions and in neurodegeneration. Previous studies have shown detrimental effects of cholesterol metabolites in neurons, but its effect in glial cells is unknown. We used a high-fat/high-cholesterol diet in mice to study the effects of hypercholesterolemia over the alarmin S100A8 cascade in the hippocampus. Using CYP27Tg, a transgenic mouse model, we show that the hypercholesterolemia influence on the brain is mediated by the excess of 27-hydroxycholesterol (27-OH), a cholesterol metabolite. We also employed an acute model of 27-OH intraventricular injection in the brain to study RAGE and S100A8 response. We used primary cultures of neurons and astrocytes to study the effect of high levels of 27-OH over the S100A8 alarmin cascade. We report that a high-fat/high-cholesterol diet leads to an increase in S100A8 production in the brain. In CYP27Tg, we report an increase of S100A8 and its receptor RAGE in the hippocampus under elevated 27-OH in the brain. Using siRNA, we found that 27-OH upregulation of RAGE in astrocytes and neurons is mediated by the nuclear receptor RXRγ. Silencing RXRγ in neurons prevented 27-OH-mediated upregulation of RAGE. These results show that S100A8 alarmin and RAGE respond to high levels of 27-OH in the brain in both neurons and astrocytes through RXRγ. Our study supports the notion that 27-OH mediates detrimental effects of hypercholesterolemia to the brain via alarmin signaling.Open access funding provided by Karolinska Institute. This research was supported by the following Swedish foundations: Swedish Brain Power, the regional agreement on medical training and clinical research (ALF) between Stockholm County Council and Karolinska Institutet, Strategic Neuroscience Program, Margaretha af Ugglas Foundation, Gun och Bertil Stohnes Stiftelse, Karolinska Institutet fund for geriatric research, Stiftelsen Gamla Tjänarinnor, Demensfonden,Lindhés Advokatbyrå, Hjärnfonden, and Alzheimerfonden. R. L.-V. was fnancially supported by Mexico’s National Council for Science and Technology (CONACYT) CVU, 209252, and by Olle Enqvist Foundation grant no. 2014/778. Ramon Areces Foundation, Spain, supported E. P., EMBO Long-Term Fellowship (ALTF 696–2013), the SSMF postdoctoral fellowship, and Juan de la Cierva-Incorporación. (IJCI-2016–27,658) supported P. M.-S

Applications of Carboxylic Acids in Organic Synthesis, Nanotechnology and Polymers

Carboxylic acids are versatile organic compounds. In this chapter is presented a current overview of the use of carboxylic acids in a different area as organic synthesis, nanotechnology, and polymers. The application carboxylic acids in these areas are: obtaining of small molecules, macromolecules, synthetic or natural polymers, modification surface of nanoparticles metallic, modification surface of nanostructure such as carbon nanotubes and graphene, nanomaterials, medical field, pharmacy, etc. Carboxylic acids can be natural and synthetic, can be extracted or synthesized, presented chemical structure highly polar, active in organic reactions, as substitution, elimination, oxidation, coupling, etc. In nanotechnology, the use of acid carboxylic as surface modifiers to promote the dispersion and incorporation of metallic nanoparticles or carbon nanostructure, in the area of polymer carboxylic acids present applications such monomers, additives, catalysts, etc. The purpose of this chapter is to emphasize the importance of carboxylic acids in different areas, highlighting the area of organic synthesis, nanotechnology and polymers and its applications

Differential Role of Type 2 Diabetes as a Risk Factor for Tuberculosis in the Elderly versus Younger Adults

The elderly are understudied despite their high risk of tuberculosis (TB). We sought to identify factors underlying the lack of an association between TB and type 2 diabetes (T2D) in the elderly, but not adults. We conducted a case–control study in elderly (≥65 years old; ELD) vs. younger adults (young/middle-aged adults (18–44/45–64 years old; YA|MAA) stratified by TB and T2D, using a research study population (n = 1160) and TB surveillance data (n = 8783). In the research study population the adjusted odds ratio (AOR) of TB in T2D was highest in young adults (AOR 6.48) but waned with age becoming non-significant in the elderly. Findings were validated using TB surveillance data. T2D in the elderly (vs. T2D in younger individuals) was characterized by better glucose control (e.g., lower hyperglycemia or HbA1c), lower insulin resistance, more sulphonylureas use, and features of less inflammation (e.g., lower obesity, neutrophils, platelets, anti-inflammatory use). We posit that differences underlying glucose dysregulation and inflammation in elderly vs. younger adults with T2D, contribute to their differential association with TB. Studies in the elderly provide valuable insights into TB-T2D pathogenesis, e.g., here we identified insulin resistance as a novel candidate mechanism by which T2D may increase active TB risk