Narrow Results

In this paper, we obtain $L^{\infty }$ estimate for the Maxwell–Higgs system in the exterior region of Reissner–Nordström spacetimes. Utilizing the integrated local energy decay estimate, the Sobolev embedding theorem alongside the Gagliardo–Nirenberg–Sobolev inequality on compact Riemannian manifold, we derive the boundedness for $L^{\infty }$ norm of the Maxwell–Higgs system on Reissner–Nordström geometry.

Extending the general undecidability result concerning the absoluteness of inequalities between subword histories, in this paper we show that the question whether such inequalities hold for all words is undecidable even over a binary alphabet and bounded number of blocks, i.e., unary factors of maximal length.

We extend and unify several Hardy type inequalities to functions whose values are positive semi-definite operators. In particular, our methods lead to the operator versions of Hardy–Hilbert's and Godunova's inequalities. While classical Hardy type inequalities hold for parameter values p > 1, it is typical that the operator versions hold only for 1 < p ≤ 2, even for functions with values in 2 × 2 matrices.

In this work, we investigate the asymptotic behavior related to the quantum privacy for multipartite systems. In this context, an inequality for quantum privacy was obtained by exploiting of quantum entropy properties. Subsequently, we derive a lower limit for the quantum privacy through the entanglement fidelity. In particular, we show that there is an interval where an increase in entanglement fidelity implies a decrease in quantum privacy.

Lower subsets of an ordered semigroup form in a natural way an ordered semigroup. This lower set operator gives an analogue of the power operator already studied in semigroup theory. We present a complete description of the lower set operator applied to varieties of ordered semigroups. We also obtain large families of fixed points for this operator applied to pseudovarieties of ordered semigroups, including all examples found in the literature. This is achieved by constructing six types of inequalities that are preserved by the lower set operator. These types of inequalities are shown to be independent in a certain sense. Several applications are also presented, including the preservation of the period for a pseudovariety of ordered semigroups whose image under the lower set operator is proper.

We investigate higher order entropies for compressible fluid models and related a priori estimates. Higher order entropies are kinetic entropy estimators suggested by Enskog expansion of Boltzmann entropy. These quantities are quadratic in the density ρ, velocity v, and temperature T renormalized derivatives. We investigate governing equations of higher order entropy correctors and related differential inequalities in the natural situation where the volume viscosity, the shear viscosity, and the thermal conductivity depend on temperature, essentially in the form T^ϰ, as given by the kinetic theory of gases. Entropic inequalities are established when ‖log ρ‖_BMO, , ‖log T‖_BMO, ‖h∂_x ρ/ρ ‖_L^∞, , ‖ h∂_xT/T‖_L^∞, and are small enough, where is a weight associated with the dependence of the local mean free path on density and temperature. As an example of application, we investigate global existence of solutions when the initial values log(ρ₀/ρ_∞), , and log(T₀/T_∞) are small enough in appropriate spaces.

In this paper, a design of strategies for players in pursuit-evasion games, is provided. The design incorporates players' preferences captured in their goal functions which are constructed using particular functional forms that include multiattribute copulas. The approach provides closed-form strategies for the players governed by nonlinear models affine in control. A number of sufficient conditions based on differential inequalities for either evasion or capture is formulated. Finally, to demonstrate effectiveness of the proposed design, some illustrative simulations with players modeled as nonlinear and nonholonomic unicycles, are provided.

Some properties of distributions f satisfying x · ∇ f ∈ L^p (ℝⁿ), 1 ≤ p < ∞, are studied. The operator x · ∇ is the generator of a semi-group of dilations. We first give Sobolev type inequalities with respect to the operator x · ∇. Using the inequalities, we also show that if , x · ∇ f ∈ L^p (ℝⁿ) and |x|^n/p|f(x)| vanishes at infinity, then f belongs to L^p (ℝⁿ). One of the Sobolev type inequalities is shown to be equivalent to the Hardy inequality in L² (ℝⁿ).

We prove that certain functions involving the gamma and q-gamma function are monotone. We also prove that (x^mψ(x))^(m+1) is completely monotonic. We conjecture that -(x^mψ^(m)(x))^(m) is completely monotonic for m ≥ 2; and we prove it, with help from Maple, for 2 ≤ m ≤ 16. We give a very useful Maple procedure to verify this for higher values of m. A stronger result is also formulated where we conjecture that the power series coefficients of a certain function are all positive.

Let n ≥ 1, x, y ∈ (0, π), and

For x ∈ [0, 1]ⁿ with ‖x‖₁ = 1 and y ∈ [1, ∞)ⁿ, we prove that

In this paper, two completely monotonic functions involving the q-gamma and the q-trigamma functions where q is a positive real, are introduced and exploited to derive sharp bounds for the q-gamma function in terms of the q-trigamma function. These results, when letting q → 1, are shown to be new. Also, sharp bounds for the q-digamma function in terms of the q-tetragamma function are derived. Furthermore, an infinite class of inequalities for the q-polygamma function is established.

The pseudo-ultraspherical polynomial of degree n can be defined by where is the ultraspherical polynomial. It is known that when λ < -n, the finite set is orthogonal on (-∞, ∞) with respect to the weight function (1 + x²)^λ-½ and when λ < 1 - n, the polynomial has exclusively real and simple zeros. Here, we undertake a deeper study of the zeros of these polynomials including bounds, numbers of real zeros, monotonicity and interlacing properties. Our methods include the Sturm comparison theorem, recurrence relations, and the explicit expression for the polynomials.

Let

In this paper, we prove formulas for the generating functions for the rank and crank differences for partitions modulo 3. In 2000, Andrews and Lewis made conjectures on inequalities satisfied by ranks and cranks modulo 3. These conjectures were first proved by Bringmann and Kane, respectively, using the circle method. Working directly on the generating functions, we obtain improvements to these inequalities.

Motivated by certain $q$-series of Ramanujan, we examine two overpartition difference functions. We give both combinatorial and asymptotic formulas for the differences and show that they are always positive. We also briefly discuss similar differences for some other types of partitions. Our main tools are elementary $q$-series transformations and Ingham’s Tauberian theorem.

In 2009, Bringmann, Lovejoy and Osburn defined two analogues of the crank function for overpartitions, namely the first residual crank and second residual crank. For a positive integer $d$, we introduce the notion of a $d$th residual crank function for overpartitions, examine the positive moments of these generalized crank functions while varying $d$, and prove some inequalities.

In this paper, we study CR-warped product submanifolds of $T$-manifolds. We prove that the CR-warped product submanifolds with invariant fiber are trivial warped products and provide a characterization theorem of CR-warped products with anti-invariant fiber of $T$-manifolds. Moreover, we develop an inequality of CR-warped product submanifolds for the second fundamental form in terms of warping function and the equality cases are considered. Also, we find a necessary and sufficient condition for compact oriented CR-warped products turning into CR-products of $T$-space forms.

In this paper, the authors prove some inequalities and completely monotonic properties of polygamma functions. As an application, we give lower bound for the zeta function on natural numbers. Partially, we answer the fifth and sixth open problems listed in [F. Qi and R. P. Agarwal, On complete monotonicity for several classes of functions related to ratios of gamma functions, J. Inequalities Appl.2019(36) (2019) 42]. We propose two open problems on completely monotonic functions related to polygamma functions.

This paper considers the well-known Erdös–Lax and Turán-type inequalities that relate the sup-norm of a univariate complex coefficient polynomial and its derivative, when there is a restriction on its zeros. The obtained results produce inequalities that are sharper than the previous ones. Moreover, a numerical example is presented, showing that in some situations, the bounds obtained by our results can be considerably sharper than the ones previously known.