Control Design for Linear Strictly Metzler Time-delay Systems

The relationships among structural constraints and involvement of the design condition are studied to synthesize state control for one class of linear strictly Metzler time-delay systems. These characterizations reflect the specific dynamical and structural attributes of the system class and outline the associated structures of linear matrix inequalities. Adjusting diagonal forms of linear matrix variables it is indicated how the proposed method gives a computable technique for the Metzler time-delay system, guaranteeing stabilising effect through implicit diagonal stabilization. The aim of this research is to describe conditions tying together inequality formulations and concepts of control theory in structures of Metzler systems

On Artin’s conjecture for pairs of diagonal forms

Let [Formula: see text] be an odd prime and [Formula: see text]. In the spirit of Artin’s conjecture, consider the system of two diagonal forms of degree [Formula: see text] in [Formula: see text] variables given by [Formula: see text] [Formula: see text] with [Formula: see text]. For [Formula: see text], this paper shows that this system has a non-trivial [Formula: see text]-adic solution for every [Formula: see text], and for every [Formula: see text], where [Formula: see text]. Moreover, for [Formula: see text], this system will have a non-trivial [Formula: see text]-adic solution for every [Formula: see text].

Importance of self-connections for brain connectivity and spectral connectomics

Spectral analysis and neural field theory are used to investigate the role of local connections in brain connectivity matrices (CMs) that quantify connectivity between pairs of discretized brain regions. This work investigates how the common procedure of omitting such self-connections (i.e., the diagonal elements of CMs) in published studies of brain connectivity affects the properties of functional CMs (fCMs) and the mutually consistent effective CMs (eCMs) that correspond to them. It is shown that retention of self-connections in the fCM calculated from two-point activity covariances is essential for the fCM to be a true covariance matrix, to enable correct inference of the direct total eCMs from the fCM, and to ensure their compatibility with it; the deCM and teCM represent the strengths of direct connections and all connections between points, respectively. When self-connections are retained, inferred eCMs are found to have net inhibitory self-connections that represent the local inhibition needed to balance excitation via white matter fibers at longer ranges. This inference of spatially unresolved connectivity exemplifies the power of spectral connectivity methods, which also enable transformation of CMs to compact diagonal forms that allow accurate approximation of the fCM and total eCM in terms of just a few modes, rather than the full $$N^2$$ N 2 CM entries for connections between N brain regions. It is found that omission of fCM self-connections affects both local and long-range connections in eCMs, so they cannot be omitted even when studying the large-scale. Moreover, retention of local connections enables inference of subgrid short-range inhibitory connectivity. The results are verified and illustrated using the NKI-Rockland dataset from the University of Southern California Multimodal Connectivity Database. Deletion of self-connections is common in the field; this does not affect case-control studies but the present results imply that such fCMs must have self-connections restored before eCMs can be inferred from them.

Diagonal forms of higher degree over function fields of p-adic curves

We investigate diagonal forms of degree [Formula: see text] over the function field [Formula: see text] of a smooth projective [Formula: see text]-adic curve: if a form is isotropic over the completion of [Formula: see text] with respect to each discrete valuation of [Formula: see text], then it is isotropic over certain fields [Formula: see text], [Formula: see text] and [Formula: see text]. These fields appear naturally when applying the methodology of patching; [Formula: see text] is the inverse limit of the finite inverse system of fields [Formula: see text]. Our observations complement some known bounds on the higher [Formula: see text]-invariant of diagonal forms of degree [Formula: see text]. We only consider diagonal forms of degree [Formula: see text] over fields of characteristic not dividing [Formula: see text].

Diophantine Inequalities for Generic Ternary Diagonal Forms

Let $k\geq 2$ and consider the Diophantine inequality $$ \left|x_1^k-{\alpha}_2 x_2^k-{\alpha}_3 x_3^k\right| <{\theta}.$$Our goal is to find non-trivial solutions in the variables $x_i$, $1\leq i\leq 3$, all of size about $P$, assuming that ${\theta }$ is sufficiently large. We study this problem on average over ${\alpha }_3$ and generalize previous work by Bourgain on quadratic ternary diagonal forms to general degree $k$.