Nonnegative Part Research Articles

Generalization of quantum calculus and corresponding Hermite–Hadamard inequalities

The aim of this paper is first to introduce generalizations of quantum integrals and derivatives which are called (ϕ-h)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\phi \\,-\\,h)$$\\end{document} integrals and (ϕ-h)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\phi \\,-\\,h)$$\\end{document} derivatives, respectively. Then we investigate some implicit integral inequalities for (ϕ-h)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\phi \\,-\\,h)$$\\end{document} integrals. Different classes of convex functions are used to prove these inequalities for symmetric functions. Under certain assumptions, Hermite–Hadamard-type inequalities for q-integrals are deduced. The results presented herein are applicable to convex, m-convex, and ħ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbar $$\\end{document}-convex functions defined on the non-negative part of the real line.

Product structure and regularity theorem for totally nonnegative flag varieties

The totally nonnegative flag variety was introduced by Lusztig. It has enriched combinatorial, geometric, and Lie-theoretic structures. In this paper, we introduce a (new) J\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$J$\\end{document}-total positivity on the full flag variety of an arbitrary Kac-Moody group, generalizing the (ordinary) total positivity.We show that the J\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$J$\\end{document}-totally nonnegative flag variety has a cellular decomposition into totally positive J\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$J$\\end{document}-Richardson varieties. Moreover, each totally positive J\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$J$\\end{document}-Richardson variety admits a favorable decomposition, called a product structure. Combined with the generalized Poincare conjecture, we prove that the closure of each totally positive J\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$J$\\end{document}-Richardson variety is a regular CW complex homeomorphic to a closed ball. Moreover, the J\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$J$\\end{document}-total positivity on the full flag provides a model for the (ordinary) totally nonnegative partial flag variety. As a consequence, we prove that the closure of each (ordinary) totally positive Richardson variety is a regular CW complex homeomorphic to a closed ball, confirming conjectures of Galashin, Karp and Lam in (Adv. Math. 351:614–620, 2019). We also show that the link of the totally nonnegative part of U−\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$U^{-}$\\end{document} for any Kac-Moody group forms a regular CW complex. This generalizes the result of Hersh (Invent. Math. 197(1):57–114, 2014) for reductive groups.

Simple Heisenberg-Virasoro modules from Weyl algebra modules

In this paper, we constructed and studied two classes of modules over the Heisenberg-Virasoro algebra HVir from Weyl algebra modules, parallel to the similar construction for Virasoro algebra in [13,9]. More precisely, given any module P over the degree-1 Weyl algebra, we define the HVir-modules M(P,V), where V is a restricted module over the nonnegative part a of HVir, and L(P,W,ξ,b,c), where W is a restricted module over another subalgebra b⊆HVir and ξ,b,c∈C with ξ≠0. We determined the irreducibility of these modules as well as the isomorphisms within each class respectively. Moreover, any two such modules from different classes are never isomorphic although they might be closely related, as we showed in the last section.

$L^p$-bounds for semigroups generated by non-elliptic quadratic differential operators

In this note, we establish L^p -bounds for the semigroup e^{-tq^w(x,D)} , t \ge 0 , generated by a quadratic differential operator q^w(x,D) on \mathbb{R}^n that is the Weyl quantization of a complex-valued quadratic form q defined on the phase space \mathbb{R}^{2n} with non-negative real part \operatorname{Re} q \ge 0 and trivial singular space. Specifically, we show that e^{-tq^w(x,D)} is bounded from L^p(\mathbb{R}^n) to L^q(\mathbb{R}^n) for all t > 0 whenever 1 \le p \le q \le \infty , and we prove bounds on \parallel { e^{-tq^w(x,D)}}_{L^p \rightarrow L^q} \parallel in both the large t \gg 1 and small 0 < t \ll 1 time regimes.Regardin g L^p \rightarrow L^q bounds for the evolution semigroup at large times, we show that \parallel{e^{-tq^w(x,D)}}_{L^p \rightarrow L^q} \parallel is exponentially decaying as t \rightarrow \infty , and we determine the precise rate of exponential decay, which is independent of (p,q) . At small times 0 < t \ll 1 , we establish bounds on \parallel {e^{-tq^w(x,D)}}_{L^p \rightarrow L^q} \parallel for (p,q) with 1 \le p \le q \le \infty that are polynomial in t^{-1} .

Shelling the \(m=1\) amplituhedron

The amplituhedron $\mathcal{A}_{n,k,m}$ was introduced by Arkani-Hamed and Trnka (2014) in order to give a geometric basis for calculating scattering amplitudes in planar $\mathcal{N}=4$ supersymmetric Yang-Mills theory. It is a projection inside the Grassmannian $\text{Gr}_{k,k+m}$ of the totally nonnegative part of $\text{Gr}_{k,n}$. Karp and Williams (2019) studied the $m=1$ amplituhedron $\mathcal{A}_{n,k,1}$, giving a regular CW decomposition of it. Its face poset $R_{n,l}$ (with $l := n-k-1$) consists of all projective sign vectors of length $n$ with exactly $l$ sign changes. We show that $R_{n,l}$ is EL-shellable, resolving a problem posed by Karp and Williams. This gives a new proof that $\mathcal{A}_{n,k,1}$ is homeomorphic to a closed ball, which was originally proved by Karp and Williams. We also give explicit formulas for the $f$-vector and $h$-vector of $R_{n,l}$, and show that it is rank-log-concave and strongly Sperner. Finally, we consider a related poset $P_{n,l}$ introduced by Machacek (2019), consisting of all projective sign vectors of length $n$ with at most $l$ sign changes. We show that it is rank-log-concave, and conjecture that it is Sperner.

Inequalities for q-h-Integrals via ℏ-Convex and m-Convex Functions

This paper investigates several integral inequalities held simultaneously for q and h-integrals in implicit form. These inequalities are established for symmetric functions using certain types of convex functions. Under certain conditions, Hadamard-type inequalities are deducible for q-integrals. All the results are applicable for ℏ-convex, m-convex and convex functions defined on the non-negative part of the real line.

On two notions of total positivity for partial flag varieties

Given integers 1≤k1<⋯<kl≤n−1, let Flk1,…,kl;n denote the type A partial flag variety consisting of all chains of subspaces (Vk1⊂⋯⊂Vkl) inside Rn, where each Vk has dimension k. Lusztig (1994, 1998) introduced the totally positive part Flk1,…,kl;n>0 as the subset of partial flags which can be represented by a totally positive n×n matrix, and defined the totally nonnegative part Flk1,…,kl;n≥0 as the closure of Flk1,…,kl;n>0. On the other hand, following Postnikov (2007), we define Flk1,…,kl;nΔ>0 and Flk1,…,kl;nΔ≥0 as the subsets of Flk1,…,kl;n where all Plücker coordinates are positive and nonnegative, respectively. It follows from the definitions that Lusztig's total positivity implies Plücker positivity, and it is natural to ask when these two notions of positivity agree. Rietsch (2009) proved that they agree in the case of the Grassmannian Flk;n, and Chevalier (2011) showed that the two notions are distinct for Fl1,3;4. We show that in general, the two notions agree if and only if k1,…,kl are consecutive integers. We give an elementary proof of this result (including for the case of Grassmannians) based on classical results in linear algebra and the theory of total positivity. We also show that the cell decomposition of Flk1,…,kl;n≥0 coincides with its matroid decomposition if and only if k1,…,kl are consecutive integers, which was previously only known for complete flag varieties, Grassmannians, and Fl1,3;4. Finally, we determine which notions of positivity are compatible with a natural action of the cyclic group of order n that rotates the index set.

Gradient Flows, Adjoint Orbits, and the Topology of Totally Nonnegative Flag Varieties

One can view a partial flag variety in \({\mathbb {C}}^n\) as an adjoint orbit \({\mathcal {O}}_{\lambda }\) inside the Lie algebra of \(n\times n\) skew-Hermitian matrices. We use the orbit context to study the totally nonnegative part of a partial flag variety from an algebraic, geometric, and dynamical perspective. The paper has three main parts: (1) We introduce the totally nonnegative part of \({\mathcal {O}}_{\lambda }\), and describe it explicitly in several cases. We define a twist map on it, which generalizes (in type A) a map of Bloch, Flaschka, and Ratiu (Duke Math. J. 61(1): 41–65, 1990) on an isospectral manifold of Jacobi matrices. (2) We study gradient flows on \({\mathcal {O}}_{\lambda }\) which preserve positivity, working in three natural Riemannian metrics. In the Kähler metric, positivity is preserved in many cases of interest, extending results of Galashin, Karp, and Lam (Adv. Math. 397: Paper No. 108123, 1–23, 2022; Adv. Math. 351: 614–620, 2019). In the normal metric, positivity is essentially never preserved on a generic orbit. In the induced metric, whether positivity is preserved appears to depends on the spacing of the eigenvalues defining the orbit. (3) We present two applications. First, we discuss the topology of totally nonnegative flag varieties and amplituhedra. Galashin, Karp, and Lam (2022, 2019) showed that the former are homeomorphic to closed balls, and we interpret their argument in the orbit framework. We also show that a new family of amplituhedra, which we call twisted Vandermonde amplituhedra, are homeomorphic to closed balls. Second, we discuss the symmetric Toda flow on \({\mathcal {O}}_{\lambda }\). We show that it preserves positivity, and that on the totally nonnegative part, it is a gradient flow in the Kähler metric up to applying the twist map. This extends a result of Bloch, Flaschka, and Ratiu (1990).

Regularity criteria for weak solutions to the 3d co-rotational Beris-Edwards system via the pressure

We investigate regularity criteria for weak solutions to the Cauchy problem of the 3d co-rotational Beris-Edwards system for nematic liquid crystals, which couples the Navier–Stokes equations for the fluid velocity u with an evolution-diffusion equations for the Q-tenser. Our results yield that for any positive constant γ>0, if either the negative part of the associated pressure Π satisfiesΠ−[ln⁡(1+Π−)]1+γ∈L∞(R+;L32,∞(R3)), or the quantity 2Π+|u|2+|∇Q|2 satisfies(2Π++|u|2+|∇Q|2)[ln⁡(1+2Π++|u|2+|∇Q|2)]1+γ∈L∞(R+;L32,∞(R3)), then the weak solution (u,Q), to the 3d co-rotational Beris-Edwards system, is global-in-time smooth. Here, the subscript “−” and “+” denote the negative and the nonnegative part, respectively. L32,∞(R3) denotes the standard weak Lebesgue space. If Q≡0, then our results extend some previous known results from the theory of the 3d Navier–Stokes equations.

Edge vectors on plabic networks in the disk and amalgamation of totally non-negative Grassmannians

The amalgamation of cluster varieties introduced by Fock and Goncharov in [21] plays a relevant role both in mathematical and physical problems. In particular, amalgamation in the totally non-negative part of positroid varieties is explicitly described as gluing of several copies of small positive Grassmannians, GrTP(1,3) and GrTP(2,3), has important topological implications [53] and naturally appears in the computation of amplitude scatterings in N=4 SYM theory [9,10]. Lam [44] has proposed to represent amalgamation in positroid varieties by equivalence classes of relations on bipartite graphs and to identify total non-negativity via appropriate edge signatures. In this paper we provide an explicit geometric characterization of such signatures in the setting of the planar bicolored trivalent directed perfect networks in the disk introduced in [52] to parametrize positroid cells SMTNN⊂GrTNN(k,n) using systems of relations for n-vectors.More precisely, to any such graph G, we associate a geometric signature satisfying both the full rank condition and the total non–negativity property on the full positroid cell. Such signature is uniquely identified by geometric indices (local winding and intersection number) ruled by the orientation O and gauge ray direction l on G.The principal result is the enrichment of Postnikov's construction in [52] by associating measurements not only to boundary edges or vertices, but to internal edges as well. Indeed we generalize prior results by Postnikov [52] and Talaska [61] providing an explicit representation of the solution to the system of geometric relations on the network (N,O,l) of graph G and positive weights. At this aim, we assign canonical basis vectors in Rn at the boundary sinks and define the vectors components at the edge e as (finite or infinite) summations over the directed paths from e to the given boundary sink. Such edge vectors have the following properties:1) They solve the geometric system of relations on (N,O,l);2) Their components are rational in the weights with subtraction–free denominators, and have explicit expressions in terms of the conservative and edge flows of [61]. At the boundary sources they coincide with the entries of the boundary measurement matrix defined in [52]. If N is acyclically orientable, all components are subtraction–free rational expressions in the weights with respect to a convenient basis. Null edge vectors may occur on reducible networks not acyclically orientable;3) We provide explicit formulas both for the transformation rules of the edge vectors with respect to the orientation and the several gauges of the given network, and for their transformations due to moves and reductions of networks.Finally, we associate a Kasteleyn orientation to the graph following [17]. If the graph is bipartite, it is known that the partition functions of dimer configurations on the graph with given boundary conditions coincide with the Plucker coordinates of the corresponding point of the totally non-negative Grassmannian [54,44,60,1,8]. In the case of plabic graphs which are not bipartite we show that the partition function for a given boundary condition is not a multiple of the corresponding minor of the boundary measurement matrix. Therefore, in this case a statistical mechanical interpretation of the boundary measurement map remains open.

Semigroups for quadratic evolution equations acting on Shubin–Sobolev and Gelfand–Shilov spaces

We consider the initial value Cauchy problem for a class of evolution equations whose Hamiltonian is the Weyl quantization of a homogeneous quadratic form with non-negative definite real part. The solution semigroup is shown to be strongly continuous on several spaces: the Shubin-Sobolev spaces, the Schwartz space, the tempered distributions, the equal index Beurling type Gelfand-Shilov spaces and their dual ultradistribution spaces.

Propagation of global analytic singularities for Schrödinger equations with quadratic Hamiltonians

We study the propagation in time of 1/2-Gelfand-Shilov singularities, i.e. global analytic singularities, of tempered distributional solutions of the initial value problem{ut+qw(x,D)u=0u|t=0=u0, on Rn, where u0 is a tempered distribution on Rn, q=q(x,ξ) is a complex-valued quadratic form on R2n=Rxn×Rξn with nonnegative real part Req≥0, and qw(x,D) is the Weyl quantization of q. We prove that the 1/2-Gelfand-Shilov singularities of the initial data that are contained within a distinguished linear subspace of the phase space R2n, called the singular space of q, are transported by the Hamilton flow of Imq, while all other 1/2-Gelfand-Shilov singularities are instantaneously regularized. Our result extends the observation of Hitrik, Pravda-Starov, and Viola '18 that this evolution is instantaneously globally analytically regularizing when the singular space of q is trivial.

Irrational Toric Varieties and Secondary Polytopes

The space of torus translations and degenerations of a projective toric variety forms a toric variety associated to the secondary fan of the integer points in the polytope corresponding to the toric variety. This is used to identify a moduli space of real degenerations with the secondary polytope. A configuration \({{\mathcal {A}}}\) of real vectors gives an irrational projective toric variety in a simplex. We identify a space of translations and degenerations of the irrational projective toric variety with the secondary polytope of \({{\mathcal {A}}}\). For this, we develop a theory of irrational toric varieties associated to arbitrary fans. When the fan is rational, the irrational toric variety is the nonnegative part of the corresponding classical toric variety. When the fan is the normal fan of a polytope, the irrational toric variety is homeomorphic to that polytope.

Necessary and sufficient conditions for extending a function to a Carathéodory function

A criterion deciding whether a function given by its values (with multiplicities) at a sequence of points in the disc $\mathbb D=\{|z|&lt;1\}$ can be extended to a holomorphic function with nonnegative real part in $\mathbb D$ is stated and proved. In the case when this function is given by the values of its derivatives at $z=0$, this is the well-known Carathéodory criterion. It is also shown that Carathéodory's criterion is a consequence of Schur's criterion and, conversely, Schur's criterion follows from Carathéodory's. Bibliography: 10 titles.

The facets of the matroid polytope and the independent set polytope of a positroid

A positroid is a special case of a realizable matroid that arose from the study of the totally nonnegative part of the Grassmannian by Postnikov. In this paper, we study the facets of its matroid polytope and the independent set polytope. This allows one to describe the bases and independent sets directly from the decorated permutation, bypassing the use of the Grassmann necklace. We also describe a criterion for determining whether a given cyclic interval is a flat or not using the decorated permutation, then show how it applies to checking the concordancy of positroids.

Reduction of the Kolmogorov inequality for a non-negative part of the second derivative on the real line to the inequality for convex functions on an interval

Reduction of the Kolmogorov inequality for a non-negative part of the second derivative on the real line to the inequality for convex functions on an interval

Regularity theorem for totally nonnegative flag varieties

We show that the totally nonnegative part of a partial flag variety G / P G/P (in the sense of Lusztig) is a regular CW complex, confirming a conjecture of Williams. In particular, the closure of each positroid cell inside the totally nonnegative Grassmannian is homeomorphic to a ball, confirming a conjecture of Postnikov.

Characterizations of Herglotz\u2013Nevanlinna Functions Using Positive Semi-Definite Functions and the Nevanlinna Kernel in Several Variables

In this paper, we give several characterizations of Herglotz–Nevanlinna functions in terms of a specific type of positive semi-definite functions called Poisson-type functions. This allows us to propose a multidimensional analogue of the classical Nevanlinna kernel and a definition of generalized Nevanlinna functions in several variables. Furthermore, a characterization of the symmetric extension of a Herglotz–Nevanlinna function is also given. The subclass of Loewner functions is discussed as well, along with an interpretation of the main result in terms of holomorphic functions on the unit polydisk with non-negative real part.

Spectral analysis of dispersive shocks for quantum hydrodynamics with nonlinear viscosity

In this paper, we investigate spectral stability of traveling wave solutions to 1D quantum hydrodynamics system with nonlinear viscosity in the [Formula: see text], that is, density and velocity, variables. We derive a sufficient condition for the stability of the essential spectrum and we estimate the maximum modulus of eigenvalues with non-negative real part. In addition, we present numerical computations of the Evans function in sufficiently large domain of the unstable half-plane and show numerically that its winding number is (approximately) zero, thus giving a numerical evidence of point spectrum stability.

Positive Configuration Space

We define and study the totally nonnegative part of the Chow quotient of the Grassmannian, or more simply the nonnegative configuration space. This space has a natural stratification by positive Chow cells, and we show that nonnegative configuration space is homeomorphic to a polytope as a stratified space. We establish bijections between positive Chow cells and the following sets: (a) regular subdivisions of the hypersimplex into positroid polytopes, (b) the set of cones in the positive tropical Grassmannian, and (c) the set of cones in the positive Dressian. Our work is motivated by connections to super Yang–Mills scattering amplitudes, which will be discussed in a sequel.