Monotonicity Inequality Research Articles

Singular Values of Dual Quaternion Matrices and Their Low-Rank Approximations

The dual quaternion matrix is an important and powerful tool for the study of multi-agent formation control. However, many basic properties of dual quaternion matrices are still not studied, due to the difficulty incurred by the non-commutativity of multiplication and existence of zero divisors of dual quaternion numbers. In this paper, we study some basic properties of dual quaternion matrices, including the polar decomposition theorem, the minimax principle and Weyl’s type monotonicity inequality for singular values, spectral norm and the Pythagoras theorem. Based upon these, we further present the best low-rank approximations for dual quaternion matrices, and discuss the relationship between the approximation degree characterizations of the best low-rank approximations in the sense of Frobenius norm and spectral norm, in addition to proving a fundamental property of the best low-rank approximation of dual quaternion matrices in a given subspace. These results are of importance to the applications in rigid body motion and data reduction.

Stochastic PDEs in 𝒮' for SDEs driven by Lévy noise

Abstract In this article we show that a finite-dimensional stochastic differential equation driven by a Lévy noise can be formulated as a stochastic partial differential equation (SPDE) driven by the same Lévy noise. We prove the existence result for such an SPDE by Itô’s formula for translation operators, and the uniqueness by an adapted form of “Monotonicity inequality”, proved earlier in the diffusion case. As a consequence, the solutions that we construct have the “translation invariance” property.

Mixed complex brightness integrals

The notion of the mixed brightness was first introduced by Lutwak. Recently, Abardia and Bernig presented complex projection bodies. Based on this notion, we define the mixed complex brightness integrals and establish related Aleksandrov–Fenchel inequality, cyclic inequality and monotonicity inequality, respectively.

Solutions of SPDE’s Associated with a Stochastic Flow

We consider the following stochastic partial differential equation, $$\begin{array}{@{}rcl@{}} &&dY_{t} = L^{\ast} Y_{t} dt + A^{\ast} Y_{t} \cdot dB_{t} &&Y_{0} = \psi, \end{array} $$ associated with a stochastic flow {X(t,x)}, for t ≥ 0, $x \in \mathbb {R}^{d}$ , as in Rajeev and Thangavelu (Potential Anal. 28(2), 139–162, 2008). We show that the strong solutions constructed there are ‘locally of compact support’. Using this notion,we define the mild solutions of the above equation and show the equivalence between strong and mild solutions in the multi Hilbertian space $\mathscr{S}^{\prime }$ . We show uniqueness of solutions in the case when ψ is smooth via the ‘monotonicity inequality’ for (L∗,A∗), which is a known criterion for uniqueness.

Dimension Bounds in Monotonicity Methods for the Helmholtz Equation

The article [HPS] established a monotonicity inequality for the Helmholtz equation and presented applications to shape detection and local uniqueness in inverse boundary problems. The monotonicity inequality states that if two scattering coefficients satisfy $q_1 \leq q_2$, then the corresponding Neumann-to-Dirichlet operators satisfy $\Lambda(q_1) \leq \Lambda(q_2)$ up to a finite dimensional subspace. Here we improve the bounds for the dimension of this space. In particular, if $q_1$ and $q_2$ have the same number of positive Neumann eigenvalues, then the finite dimensional space is trivial.

Dual mixed complex brightness integrals

In this paper, we define the dual mixed complex brightness integrals and establish related Brunn-Minkowski type inequality, Aleksandrov-Fenchel inequality, cyclic inequality and monotonicity inequality, respectively. As applications, we give the analogous version of the differences inequalities for the dual mixed complex brightness integrals

Quantum f-divergences in von Neumann algebras. II. Maximal f-divergences

As a continuation of the study by Hiai [J. Math. Phys. 59, 102202 (2018)] on standard f-divergences, we make a systematic study of maximal f-divergences in general von Neumann algebras. For maximal f-divergences, apart from their definition based on Haagerup’s L1-space, we present the general integral expression and the variational expression in terms of reverse tests. From these expressions as well as the definition, we prove important properties of maximal f-divergences, for instance, the monotonicity inequality, the joint convexity, the lower semicontinuity, and the martingale convergence. The inequality between the standard and the maximal f-divergences is also given.

Constraining quantum fields using modular theory

Tomita-Takesaki modular theory provides a set of algebraic tools in quantum field theory that is suitable for the study of the information-theoretic properties of states. For every open set in spacetime and choice of two states, the modular theory defines a positive operator known as the relative modular operator that decreases monotonically under restriction to subregions. We study the consequences of this operator monotonicity inequality for correlation functions in quantum field theory. We do so by constructing a one-parameter Rényi family of information-theoretic measures from the relative modular operator that inherit monotonicity by construction and reduce to correlation functions in special cases. In the case of finite quantum systems, this Rényi family is the sandwiched Rényi divergence and we obtain a simple proof of its monotonicity. Its monotonicity implies a class of constraints on correlation functions in quantum field theory, only a small set of which were known to us. We explore these inequalities for free fields and conformal field theory. We conjecture that the second null derivative of Rényi divergence is non-negative which is a generalization of the quantum null energy condition to the Rényi family.

Stable Big Bang formation in near-FLRW solutions to the Einstein-scalar field and Einstein-stiff fluid systems

We prove a stable blowup result for solutions to the Einstein-scalar field and Einstein-stiff fluid systems. Our result applies to small perturbations of the spatially flat Friedmann–Lemaitre–Robertson–Walker (FLRW) solution with topology $$(0,\infty ) \times \mathbb {T}^3$$ . The FLRW solution models a spatially uniform scalar-field/stiff fluid evolving in a spacetime that expands towards the future and that has a “Big Bang” singularity at $$\lbrace 0 \rbrace \times \mathbb {T}^3$$ , where various curvature invariants blow up. We place “initial” data on a Cauchy hypersurface $$\Sigma _1'$$ that are close, as measured by a Sobolev norm, to the FLRW data induced on $$\lbrace 1 \rbrace \times \mathbb {T}^3$$ . We then study the asymptotic behavior of the perturbed solution in the collapsing direction and prove that its basic qualitative and quantitative features closely resemble those of the FLRW solution. In particular, for the perturbed solution, we construct constant mean curvature-transported spatial coordinates covering $$(t,x) \in (0,1] \times \mathbb {T}^3$$ and show that it also has a Big Bang at $$\lbrace 0 \rbrace \times \mathbb {T}^3$$ , where its curvature blows up. The blowup confirms Penrose’s Strong Cosmic Censorship hypothesis for the “past-half” of near-FLRW solutions. Furthermore, we show that the equations are dominated by kinetic (that is, time-derivative-involving) terms that induce approximately monotonic behavior near the Big Bang. As a consequence of the monotonicity, we also show that various time-rescaled components of the solution converge to regular functions of x as $$t \downarrow 0$$ . The most difficult aspect of the proof is showing that the solution exists for $$(t,x) \in (0,1] \times \mathbb {T}^3$$ , and to this end, we derive a hierarchy of energy estimates that allow for the possibility of mild energy blowup as $$t \downarrow 0$$ . To close these estimates, it is essential that we are able to rule out more singular energy blowup, a step that is in turn tied to the most important ingredient in our analysis: a collection of integral identities that, when combined in the right proportions, yield an $$L^2$$ -type approximate monotonicity inequality, a key point being that the error terms are controllable up to the singularity for near-FLRW solutions. In a companion article, we derived similar approximate monotonicity inequalities for linearized versions of the Einstein-scalar field equations and used them to prove linear stability results for a family of spatially homogeneous background solutions. The present article shows that the linear stability of the FLRW background solution can be upgraded to a full proof of the nonlinear stability of its singularity.

Local energy inequalities for mean curvature flow into evolving ambient spaces

We establish a local monotonicity formula for mean curvature flow into a curved space whose metric is also permitted to evolve simultaneously with the flow, extending the work of Ecker (Ann Math (2) 154(2):503–525, 2001), Huisken (J Differ Geom 31(1):285–299, 1990), Lott (Commun Math Phys 313(2):517–533, 2012), Magni, Mantegazza and Tsatis (J Evol Equ 13(3):561–576, 2013) and Ecker et al. (J Reine Angew Math 616:89–130, 2008). This formula gives rise to a monotonicity inequality in the case where the target manifold’s geometry is suitably controlled, as well as in the case of a gradient shrinking Ricci soliton. Along the way, we establish suitable local energy inequalities to deduce the finiteness of the local monotone quantity.

A boundary monotonicity inequality for variationally biharmonic maps and applications to regularity theory

We derive a boundary monotonicity formula for a class of biharmonic maps with Dirichlet boundary conditions. A monotonicity formula is crucial in the theory of partial regularity in supercritical dimensions. As a consequence of such a boundary monotonicity formula, one is able to show partial regularity for variationally biharmonic maps and full boundary regularity for minimizing biharmonic maps.

Dual Orlicz affine surface area

According to the notion of Orlicz mixed volume, in this paper, we extend Lp-dual affine surface area to the Orlicz version. Further, we obtain the affine isoperimetric inequality and the Blachke-Santalo inequality for the dual Orlicz affine surface area. Besides, we also get the monotonicity inequality for Orlicz dual affine surface area.

Characterizing Gaussian Flows Arising from Itô’s Stochastic Differential Equations

We introduce and characterize a class of flows, which turn out to be Gaussian. This characterization allows us to show, using the Monotonicity inequality, that the transpose of the flow, for an extended class of initial conditions, is the unique solution of the SPDE introduced in Rajeev and Thangavelu (2008).

A Strengthened Monotonicity Inequality of Quantum Relative Entropy: A Unifying Approach Via Rényi Relative Entropy

We derive a strengthened monotonicity inequality for quantum relative entropy by employing properties of $\alpha$-R\'{e}nyi relative entropy. We develop a unifying treatment towards the improvement of some quantum entropy inequalities. In particular, an emphasis is put on a lower bound of quantum conditional mutual information (QCMI) as it gives a Pinsker-like lower bound for the QCMI. We also give some improved entropy inequalities based on R\'{e}nyi relative entropy. The inequalities obtained, thus, extends some well-known ones. We also obtain a condition under which a tripartite operator becomes a Markov state. As a by-product we provide some trace inequalities of operators, which are of independent interest.

Dual Orlicz geominimal surface area

The $L_{p}$ -geominimal surface area was introduced by Lutwak in 1996, which extended the important concept of the geominimal surface area. Recently, Wang and Qi defined the p-dual geominimal surface area, which belongs to the dual Brunn-Minkowski theory. In this paper, based on the concept of the dual Orlicz mixed volume, we extend the dual geominimal surface area to the Orlicz version and give its properties. In addition, the isoperimetric inequality, a Blaschke-Santalo type inequality, and the monotonicity inequality for the dual Orlicz geominimal surface areas are established.

Enclosure method for the p-Laplace equation

We study the enclosure method for the p-Calderón problem, which is a nonlinear generalization of the inverse conductivity problem due to Calderón that involves the p-Laplace equation. The method allows one to reconstruct the convex hull of an inclusion in the nonlinear model by using exponentially growing solutions introduced by Wolff. We justify this method for the penetrable obstacle case, where the inclusion is modelled as a jump in the conductivity. The result is based on a monotonicity inequality and the properties of the Wolff solutions.

Differential operators on Hermite Sobolev spaces

In this paper, we compute the Hilbert space adjoint ∂∗ of the derivative operator ∂ on the Hermite Sobolev spaces \(\mathcal {S}_{q}\). We use this calculation to give a different proof of the ‘monotonicity inequality’ for a class of differential operators (L,A) for which the inequality was proved in Infin. Dimens. Anal. Quantum Probab. Relat. Top.2(4) (2009) 515–591. We also prove the monotonicity inequality for (L,A), when these correspond to the Ornstein–Uhlenbeck diffusion.

Weak monotonicity and Chebyshev type inequality

Weak monotonicity and Chebyshev type inequality

Inequalities for interpolation functions

In this paper, in relation with interpolation functions we study some generalized Powers-Stormer's type inequalities and monotonicity inequality of indefinite type which generalizes a result of Ando.

ON SOME PROPERTIES OF TWO VECTOR-VALUED VAR AND CTE MULTIVARIATE RISK MEASURES FOR ARCHIMEDEAN COPULAS

AbstractWe consider the multivariate Value-at-Risk (VaR) and Conditional-Tail-Expectation (CTE) risk measures introduced in Cousin and Di Bernardino (Cousin, A. andDi Bernardino, E. (2013)Journal of Multivariate Analysis,119, 32–46;Cousin, A. andDi Bernardino, E. (2014)Insurance: Mathematics and Economics,55(C), 272–282). For absolutely continuous Archimedean copulas, we derive integral formulas for the multivariate VaR and CTE Archimedean risk measures. We show that each component of the multivariate VaR and CTE functional vectors is an integral transform of the corresponding univariate VaR measures. For the class of Archimedean copulas, the marginal components of the CTE vector satisfy the following properties: positive homogeneity (PH), translation invariance (TI), monotonicity (MO), safety loading (SL) and VaR inequality (VIA). In case marginal risks satisfy the subadditivity (MSA) property, the marginal CTE components are also sub-additive and hitherto coherent risk measures in the usual sense. Moreover, the increasing risk (IR) or stop-loss order preserving property of the marginal CTE components holds for the class of bivariate Archimedean copulas. A counterexample to the (IR) property for the trivariate Clayton copula is included.