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The administration of goat milk fermented (FGM) with Lactobacillus delbrueckii subsp. indicus CRL1447 and supplemented with different mixes of lactobacilli strains (Mix1: Limosilactobacillus fermentum CRL1446 + Lactiplantibacillus paraplantarum CRL1449 + Lactiplantibacillus paraplantarum CRL1472; Mix2: CRL1446 + CRL1449; Mix3: CRL1446 + CRL1472; and Mix4: CRL1449 + CRL1472) was investigated regarding body weight, metabolic and inflammatory parameters, and gut microbiota (GM) composition in mice fed a high-fat diet (HFD). Body weight gain, adipocyte size, fasting blood glucose, serum triglyceride, and leptin levels were significantly reduced in the group fed FGM+Mix3 compared with the obese mice fed FGM. FGM+Mix2 and FGM+Mix3 modified the GM composition, reversing the dysbiosis caused by the HFD. Although there were no significant changes at the phylum level, the GM composition was significantly changed at the family and genus levels. Results suggest that the administration of FGM+Mix3 improves metabolic and immune profiles in obese mice while positively modulating the GM, therefore attenuating the risk factors associated with obesity. [ABSTRACT FROM AUTHOR]

We describe an atlas adapted to the Toda flow on the manifold of full flags of any non-compact real semisimple Lie algebra and on its Hessenberg-type submanifolds. We show that in the local coordinates of the atlas the Toda flow becomes linear. The local coordinates are also used to show that the Toda flow on the manifold of full flags is Morse–Smale, which generalizes the result for traceless matrices in [ 27 ] to arbitrary non-compact real semisimple Lie algebras. As a byproduct we describe new features of classical constructions in matrix theory. [ABSTRACT FROM AUTHOR]

Originally a model for wave propagation on the line, the Toda lattice is a wonderful case study in mechanics and symplectic geometry. In Flaschka's variables, it becomes an evolution given by a Lax pair on the vector space of real, symmetric, tridiagonal matrices. Its very special asymptotic behavior was studied by Moser by introducing norming constants, which play the role of discrete inverse variables in analogy to the solution by inverse scattering of KdV. It is a completely integrable system on the coadjoint orbit of the upper triangular group. Recently, bidiagonal coordinates, which parameterize also non-Jacobi tridiagonal matrices, were used to reduce asymptotic questions to local theory. Larger phase spaces for the Toda lattice lead to the study of isospectral manifolds and different coadjoint orbits. Additionally, the time one map of the associated flow is computed by a familiar algorithm in numerical linear algebra. The text is mostly expositive and self contained, presenting alternative formulations of familiar results and applications to numerical analysis. [ABSTRACT FROM AUTHOR]

A common algorithm for the computation of eigenvalues of real symmetric tridiagonal matrices is the iteration of certain special maps Fσ called shifted QR steps. Such maps preserve spectrum and a natural common domain is , the manifold of real symmetric tridiagonal matrices conjugate to the diagonal matrix Λ. More precisely, a (generic) shift defines a map . A strategy specifies the shift to be applied at T so that Fσ(T)=Fσ(T)(T). Good shift strategies should lead to fast deflation: some off-diagonal coordinate tends to zero, allowing for reduction of the problem to submatrices. For topological reasons, continuous shift strategies do not obtain fast deflation; many standard strategies are indeed discontinuous. Practical implementation only gives rise systematically to bottom deflation, convergence to zero of the lowest off-diagonal entry b(T). For most shift strategies, convergence to zero of b(T) is cubic, |b(Fσ(T))|=Θ(|b(T)|k) for k=3. The existence of arithmetic progressions in the spectrum of T sometimes implies instead quadratic convergence, k=2. The complete integrability of the Toda lattice and the dynamics at non-smooth points are central to our discussion. The text does not assume knowledge of numerical linear algebra. [ABSTRACT FROM PUBLISHER]

Discretizations by finite differences of some semilinear elliptic equations lead to maps F(u) = Au - f (u), u ∈ …n, given by nonlinear convex diagonal perturbations of symmetric matrices A. For natural nonlinearity classes, we consider the equation F(u) = y - tp, where t is a large positive number and p is a vector with negative coordinates. As the range of the derivative f′i of the coordinates of f encloses more eigenvalues of A, the number of solutions increases geometrically, eventually reaching 2n. This phenomenon, somewhat in contrast with behaviour associated with the Lazer-McKenna conjecture, has a very simple geometric explanation: a perturbation of a multiple fold gives rise to a function which sends connected components of its critical set to hypersurfaces with large rotation numbers with respect to vectors with very negative coordinates. Strictly speaking, the results have nothing to do with elliptic equations: they are properties of the interaction of a (self-adjoint) linear map with increasingly stronger nonlinear convex diagonal interactions. [ABSTRACT FROM AUTHOR]

One of the most widely used methods for eigenvalue computation is the QR iteration with Wilkinson’s shift: Here, the shift s is the eigenvalue of the bottom 2×2 principal minor closest to the corner entry. It has been a long-standing question whether the rate of convergence of the algorithm is always cubic. In contrast, we show that there exist matrices for which the rate of convergence is strictly quadratic. More precisely, let $T_{ {\mathcal {X}}}$ be the 3×3 matrix having only two nonzero entries $(T_{ {\mathcal {X}}})_{12}=(T_{ {\mathcal {X}}})_{21}=1$ and let ${\mathcal {T}}_{\varLambda }$ be the set of real, symmetric tridiagonal matrices with the same spectrum as $T_{ {\mathcal {X}}}$ . There exists a neighborhood $\boldsymbol {{\mathcal {U}}}\subset {\mathcal {T}}_{\varLambda }$ of $T_{ {\mathcal {X}}}$ which is invariant under Wilkinson’s shift strategy with the following properties. For $T_{0}\in \boldsymbol {{\mathcal {U}}}$ , the sequence of iterates ( T k) exhibits either strictly quadratic or strictly cubic convergence to zero of the entry ( T k)23. In fact, quadratic convergence occurs exactly when $\lim T_{k}=T_{ {\mathcal {X}}}$ . Let $\boldsymbol {{\mathcal {X}}}$ be the union of such quadratically convergent sequences ( T k): The set $\boldsymbol {{\mathcal {X}}}$ has Hausdorff dimension 1 and is a union of disjoint arcs $\boldsymbol {{\mathcal {X}}}^{\sigma}$ meeting at $T_{ {\mathcal {X}}}$ , where σ ranges over a Cantor set. [ABSTRACT FROM AUTHOR]

In this paper we compute the spectra with multiplicities of the adjancency graphs of all regular polytopes in ℝ: it turns out that all such eigenvalues are algebraic intergers of degree no greater than 3. [ABSTRACT FROM AUTHOR]

We compute the spectra of the adjacency matrices of the semi-regular polytopes. A few different techniques are employed: the most sophisticated, which relates the 1-skeleton of the polytope to a Cayley graph, is based on methods akin to those of Lovász and Babai ([L], [B]). It turns out that the algebraic degree of the eigenvalues is at most 5, achieved at two 3-dimensional solids. [ABSTRACT FROM AUTHOR]