Instability, index theorem, and exponential trichotomy for Linear Hamiltonian PDEs

Consider a general linear Hamiltonian system ∂ t u = J L u \partial _{t}u=JLu in a Hilbert space X X . We assume that L : X → X ∗ \ L:X\rightarrow X^{\ast } induces a bounded and symmetric bi-linear form ⟨ L ⋅ , ⋅ ⟩ \left \langle L\cdot ,\cdot \right \rangle on X X , which has only finitely many negative dimensions n − ( L ) n^{-}(L) . There is no restriction on the anti-self-dual operator J : X ∗ ⊃ D ( J ) → X J:X^{\ast }\supset D(J)\rightarrow X . We first obtain a structural decomposition of X X into the direct sum of several closed subspaces so that L L is blockwise diagonalized and J L JL is of upper triangular form, where the blocks are easier to handle. Based on this structure, we first prove the linear exponential trichotomy of e t J L e^{tJL} . In particular, e t J L e^{tJL} has at most algebraic growth in the finite co-dimensional center subspace. Next we prove an instability index theorem to relate n − ( L ) n^{-}\left ( L\right ) and the dimensions of generalized eigenspaces of eigenvalues of J L \ JL , some of which may be embedded in the continuous spectrum. This generalizes and refines previous results, where mostly J J was assumed to have a bounded inverse. More explicit information for the indexes with pure imaginary eigenvalues are obtained as well. Moreover, when Hamiltonian perturbations are considered, we give a sharp condition for the structural instability regarding the generation of unstable spectrum from the imaginary axis. Finally, we discuss Hamiltonian PDEs including dispersive long wave models (BBM, KDV and good Boussinesq equations), 2D Euler equation for ideal fluids, and 2D nonlinear Schrödinger equations with nonzero conditions at infinity, where our general theory applies to yield stability or instability of some coherent states.

A study on product-sum of triangular fuzzy numbers

We study the problem: if a˜i, i ∈ N are fuzzy numbers of triangular form, then what is the membership function of the infinite (or finite) sum -˜a1 + a˜2 + · · · (defined via the sub-product-norm convolution)

On the semi-scalar equivalence of polynomial matrices over a field

Polynomial matrices and of size over a field are semi-scalar equivalent if there exist a nonsingular matrix over and an invertible matrix over such that . The aim of the present report is to present a triangular form of some nonsingular polynomial matrices with respect to semi-scalar equivalence.

Predicting the shape of the obturator foramen in the surgical treatment of stress urinary incontinence in women

The aim of the study was to study the possibility of predicting the shape of the obturator foramen depending on the shape of the pelvic bone cavity in the aspect of the surgical treatment of stress urinary incontinence in adult women. Materials and methods. 61 preparations of the female bone pelvis were studied. A pelviometric form was developed, according to which the linear and angular parameters of the pelvis and obturator foramen, indices of the pelvic cavity and obturator foramen were evaluated. A discriminant analysis was applied to classify the shapes of the obturator foramen. Results. Based on the calculated pelvic cavity indices, the shape of the pelvic cavity was determined: narrowing to the bottom, cylindrical and widening to the bottom. Using discriminant analysis, a model was developed to predict the shape of the obturator foramen, depending on the shape of the pelvis. Two main forms of the obturator foramen are distinguished: triangular and elliptical. It was found that the triangular form of the obturator foramen is most characteristic of the cylindrical form of the pelvic cavity and to a lesser extent of the downwardly tapering form. The ellipsoidal shape of the obturator foramen predominated in the downwardly expanding pelvic cavity. It is also shown that a narrow under-pelvic angle is characteristic of the downward pelvic cavity and a wide under-pelvic angle is characteristic of the downward dilating pelvis. Conclusion. The presented index of the lateral deviation of the ischial tubercles makes it possible to determine the shape of the pelvic cavity: narrowing to the bottom, cylindrical and dilating to the bottom. The method of discriminant analysis provides a high degree of certainty in predicting the shape of the obturator foramen, depending on the shape of the pelvic cavity.

The influence of the charge shape on heat exchange of a recessed nozzle

The paper deals with the numerical simulation of the flow of thermally conductive viscous gaseous combustion products in the flow paths of a power plant. The influence of the shape of the mass supply surface on the gas dynamics and heat exchange near the recessed nozzle of the power plant is investigated. The coupled problem of heat exchange is solved by the method of control volumes. It is shown that the compensator geometry determines the localization of both the topological features of the flow near the recessed nozzle and the position of local maximums of the heat transfer coefficient. It has been revealed that The use of a channel with a star-shaped cross section and a triangular form of compensator rays leads to an intensification of heat exchange processes near a recessed nozzle.

The Use of Solvable Directed Graphs in a Jacobi-like Algorithm

In this paper, we introduce a Jacobi-like algorithm (we call D-NJLA) to reduce a real nonsymmetric n × n matrix to a real upper triangular form by the help of solvable directed graphs. This method uses only real arithmetic and a sequence of orthogonal similarity transformations and achieves ultimate quadratic convergence. A theoretical analysis is constructed and some experimental results are given.

Twin modular S4 with SU(5) GUT

We discuss the SU(5) grand unified extension of flavour models with multiple modular symmetries. The proposed model involves two modular S4 groups, one acting in the charged fermion sector, associated with a modulus field value τT with residual $$ {Z}_3^T $$ Z 3 T symmetry, and one acting in the right-handed neutrino sector, associated with another modulus field value τSU with residual $$ {Z}_2^{SU} $$ Z 2 SU symmetry. Quark and lepton mass hierarchies are naturally generated with the help of weightons, which are SM singlet fields, where their non-zero modular weights play the role of Froggatt-Nielsen charges. The model predicts TM1 lepton mixing, and neutrinoless double beta decay at rates close to the sensitivity of current and future experiments, for both normal and inverted orderings, with suppressed corrections from charged lepton mixing due to the triangular form of its Yukawa matrix.