GENERALIZATION AND REFINEMENTS OF JENSEN INEQUALITY

We give generalizations and refinements of Jensen and Jensen− Mercer inequalities by using weights which satisfy the conditions of Jensen and Jensen− Steffensen inequalities. We also give some refinements for discrete and integral version of generalized Jensen−Mercer inequality and shown to be an improvement of the upper bound for the Jensen’s difference given in [32]. Applications of our work include new bounds for some important inequalities used in information theory, and generalizing the relations among means.

ISS and integral-ISS of switched systems with nonlinear supply functions

The problem of input-to-state stability (ISS) and its integral version (iISS) are considered for switched nonlinear systems with inputs, resets and possibly unstable subsystems. For the dissipation inequalities associated with the Lyapunov function of each subsystem, it is assumed that the supply functions, which characterize the decay rate and ISS/iISS gains of the subsystems, are nonlinear. The change in the value of Lyapunov functions at switching instants is described by a sum of growth and gain functions, which are also nonlinear. Using the notion of average dwell-time (ADT) to limit the number of switching instants on an interval, and the notion of average activation time (AAT) to limit the activation time for unstable systems, a formula relating ADT and AAT is derived to guarantee ISS/iISS of the switched system. Case studies of switched systems with saturating dynamics and switched bilinear systems are included for illustration of the results.

Bounds for the Jensen Gap in terms of Power Means with Applications

Jensen’s and its related inequalities have attracted the attention of several mathematicians due to the fact that Jensen’s inequality has numerous applications in almost all disciplines of mathematics and in other fields of science. In this article, we propose new bounds for the difference of two sides of Jensen’s inequality in terms of power means. An example has been presented for the importance and support of the main results. Related results have been given in quantum calculus. As consequences, improvements of quantum integral version of Hermite-Hadamard inequality have been derived. The obtained inequalities have been applied for some well-known inequalities such as Hermite-Hadamrd, Hölder, and power mean inequalities. Finally, some applications are given in information theory. The tools performed for obtaining the main results may be applied to obtain more results for other inequalities.

A geometric Jacquet–Langlands correspondence for paramodular Siegel threefolds

We study the Picard–Lefschetz formula for Siegel modular threefolds of paramodular level and prove the weight-monodromy conjecture for its middle degree inner cohomology. We give some applications to the Langlands programme: using Rapoport-Zink uniformisation of the supersingular locus of the special fiber, we construct a geometric Jacquet–Langlands correspondence between $${\text {GSp}}_4$$ GSp 4 and a definite inner form, proving a conjecture of Ibukiyama. We also prove an integral version of the weight-monodromy conjecture and use it to deduce a level lowering result for cohomological cuspidal automorphic representations of $${\text {GSp}}_4$$ GSp 4 .

Uniform, Integral, and Feasible Proofs for the Determinant Identities

Aiming to provide weak as possible axiomatic assumptions in which one can develop basic linear algebra, we give a uniform and integral version of the short propositional proofs for the determinant identities demonstrated over GF (2) in Hrubeš-Tzameret [15]. Specifically, we show that the multiplicativity of the determinant function and the Cayley-Hamilton theorem over the integers are provable in the bounded arithmetic theory VNC 2 ; the latter is a first-order theory corresponding to the complexity class NC 2 consisting of problems solvable by uniform families of polynomial-size circuits and O (log 2 n )-depth. This also establishes the existence of uniform polynomial-size propositional proofs operating with NC 2 -circuits of the basic determinant identities over the integers (previous propositional proofs hold only over the two-element field).

A strictly commutative model for the cochain algebra of a space

The commutative differential graded algebra $A_{\mathrm {PL}}(X)$ of polynomial forms on a simplicial set $X$ is a crucial tool in rational homotopy theory. In this note, we construct an integral version $A^{\mathcal {I}}(X)$ of $A_{\mathrm {PL}}(X)$. Our approach uses diagrams of chain complexes indexed by the category of finite sets and injections $\mathcal {I}$ to model $E_{\infty }$ differential graded algebras (dga) by strictly commutative objects, called commutative $\mathcal {I}$-dgas. We define a functor $A^{\mathcal {I}}$ from simplicial sets to commutative $\mathcal {I}$-dgas and show that it is a commutative lift of the usual cochain algebra functor. In particular, it gives rise to a new construction of the $E_{\infty }$ dga of cochains. The functor $A^{\mathcal {I}}$ shares many properties of $A_{\mathrm {PL}}$, and can be viewed as a generalization of $A_{\mathrm {PL}}$ that works over arbitrary commutative ground rings. Working over the integers, a theorem by Mandell implies that $A^{\mathcal {I}}(X)$ determines the homotopy type of $X$ when $X$ is a nilpotent space of finite type.

Trajectory tracking for quadrotor UAV transporting cable-suspended payload in wind presence

In this paper, trajectory tracking for a quadrotor unmanned aerial vehicle (UAV) is considered in the presence of a cable-suspended payload and wind as unknown disturbances. It is assumed that the wind disturbance is slowly time varying and affects quadrotor position and orientation independently. Nonlinear robust strategies, such as backstepping and sliding mode control could be used for trajectory tracking; however, they fail to stabilize the system in the presence of payload or wind, or both together. We have proposed a combined backstepping and super-twisting integral sliding mode strategy to stabilize the system. Conventional sliding mode control suggests discontinuous control signals and suffers from the chattering phenomenon whereas its super-twisting integral version suggests continuous control signals, which makes it implementable and chattering free.

Generalized Jensen-Mercer Inequality for Functions with Nondecreasing Increments

In the year 2003, McD Mercer established an interesting variation of Jensen’s inequality and later in 2009 Mercer’s result was generalized to higher dimensions by M. Niezgoda. Recently, Asif et al. has stated an integral version of Niezgoda’s result for convex functions. We further generalize Niezgoda’s integral result for functions with nondecreasing increments and give some refinements with applications. In the way, we generalize an important result, Jensen-Boas inequality, using functions with nondecreasing increments. These results would constitute a valuable addition to Jensen-type inequalities in the literature.

CANONICAL REPRESENTATIVES FOR DIVISOR CLASSES ON TROPICAL CURVES AND THE MATRIX–TREE THEOREM

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\Gamma $ be a compact tropical curve (or metric graph) of genus $g$. Using the theory of tropical theta functions, Mikhalkin and Zharkov proved that there is a canonical effective representative (called a break divisor) for each linear equivalence class of divisors of degree $g$ on $\Gamma $. We present a new combinatorial proof of the fact that there is a unique break divisor in each equivalence class, establishing in the process an ‘integral’ version of this result which is of independent interest. As an application, we provide a ‘geometric proof’ of (a dual version of) Kirchhoff’s celebrated matrix–tree theorem. Indeed, we show that each weighted graph model $G$ for $\Gamma $ gives rise to a canonical polyhedral decomposition of the $g$-dimensional real torus $\mathrm{Pic}^g(\Gamma )$ into parallelotopes $C_T$, one for each spanning tree $T$ of $G$, and the dual Kirchhoff theorem becomes the statement that the volume of $\mathrm{Pic}^g(\Gamma )$ is the sum of the volumes of the cells in the decomposition.