GIT Stability Research Articles

Polymeric Nanoparticles for Delivery of Natural Bioactive Agents: Recent Advances and Challenges.

In the last few decades, several natural bioactive agents have been widely utilized in the treatment and prevention of many diseases owing to their unique and versatile therapeutic effects, including antioxidant, anti-inflammatory, anticancer, and neuroprotective action. However, their poor aqueous solubility, poor bioavailability, low GIT stability, extensive metabolism as well as short duration of action are the most shortfalls hampering their biomedical/pharmaceutical applications. Different drug delivery platforms have developed in this regard, and a captivating tool of this has been the fabrication of nanocarriers. In particular, polymeric nanoparticles were reported to offer proficient delivery of various natural bioactive agents with good entrapment potential and stability, an efficiently controlled release, improved bioavailability, and fascinating therapeutic efficacy. In addition, surface decoration and polymer functionalization have opened the door to improving the characteristics of polymeric nanoparticles and alleviating the reported toxicity. Herein, a review of the state of knowledge on polymeric nanoparticles loaded with natural bioactive agents is presented. The review focuses on frequently used polymeric materials and their corresponding methods of fabrication, the needs of such systems for natural bioactive agents, polymeric nanoparticles loaded with natural bioactive agents in the literature, and the potential role of polymer functionalization, hybrid systems, and stimuli-responsive systems in overcoming most of the system drawbacks. This exploration may offer a thorough idea of viewing the polymeric nanoparticles as a potential candidate for the delivery of natural bioactive agents as well as the challenges and the combating tools used to overcome any hurdles.

On K‐stability of some del Pezzo surfaces of Fano index 2

For every integer $a \geq 2$, we relate the K-stability of hypersurfaces in the weighted projective space $\mathbb{P}(1,1,a,a)$ of degree $2a$ with the GIT stability of binary forms of degree $2a$. Moreover, we prove that such a hypersurface is K-polystable and not K-stable if it is quasi-smooth.

Gravitating vortices with positive curvature

We give a complete solution to the existence problem for gravitating vortices with non-negative topological constant c⩾0. Our first main result builds on previous results by Yang and establishes the existence of solutions to the Einstein-Bogomol'nyi equations, corresponding to c=0, in all admissible Kähler classes. Our second main result completely solves the existence problem for c>0. Both results are proved by the continuity method and require that a GIT stability condition for an effective divisor on the Riemann sphere is satisfied. For the former, the continuity path starts from a given solution with c=0 and deforms the Kähler class. For the latter result we start from the established solution in any fixed admissible Kähler class and deform the coupling constant α towards 0. A salient feature of our argument is a new bound Sg⩾c for the curvature of gravitating vortices, which we apply to construct a limiting solution along the path via Cheeger-Gromov theory.

Stable linear systems of skew-symmetric forms of generic rank ≤4

Given a 6-dimensional complex vector space W, we consider linear systems of skew-symmetric forms on W. The n-dimensional linear systems of this kind, that can also be interpreted as n-dimensional linear subspaces of P(⋀2W⁎), are parametrized by the projective space P(Cn+1⊗⋀2W⁎). We analyse the SL(W) action on this projective space and the GIT stability of linear systems with respect to this action. We present a classification of all stable orbits of linear systems whose generic element is a tensor of rank 4.

Design and development of lipid modified chitosan containing muco-adhesive self-emulsifying drug delivery systems for cefixime oral delivery

Current study was aimed to design and develop muco-adhesive self-nano emulsifying drug delivery system (SNEDDs) for improved pharmacokinetics of Cefixime (CFX) in rabbits. The components of SNEDDs formulation i.e., cinnamon oil, Tween® 80, and PEG 200 as oil, surfactant, and co-surfactant respectively were selected based on their high solubilizing capability of the drug. SNEDDs formulation was optimized using Design of experiments (D-optimal design) in terms of droplet size, poly dispersity index and zeta potential. The optimized SNEDDs formulation was studied for various parameters like droplet size, morphology, zeta potential, emulsification, optical clarity, thermodynamic stability, GIT stability, and robustness to dilution. CFX was loaded to optimized formulation to form CFX-SNEDDs. Furthermore, acyl-chitosan, a muco-adhesive agent, was added to CFX-SNEDDS to prepare CHT-CFX-SNEDDS. In vitro drug release showed the controlled release behavior reached a maximum value of 70 % at pH 6.8 within 24 h. The droplet size, atomic force microscopy, and optical clarity analysis revealed the formation of nanosized emulsion (156 ± 25 nm) with spherical morphology. Also in vivo pharmacokinetic studies on rabbits showed an increased drug plasma concentration for CHT-CFX-SNEDDs (15 ± 3 μg/mL) and CFX-SNEDDs (9 ± 2 μg/mL) in comparison with control CFX (4 ± 1 μg/mL). The results indicated that the developed CHT-CFX-SNEDDs with an increased degree of solubilization, permeation, and nanosized range emulsion enhance the oral performance of CFX.

Encapsulation of lipophilic polyphenols in plant-based nanoemulsions: impact of carrier oil on lipid digestion and curcumin, resveratrol and quercetin bioaccessibility.

Lipophilic polyphenol compounds (LPCs) are claimed to exhibit a broad spectrum of biological activities that may improve human health and wellbeing, including antioxidant, anti-inflammatory, and anti-cancer properties. Nanoemulsion-based delivery systems have been developed to encapsulate LPCs so as to increase their food matrix compatibility, physicochemical stability, and bioavailability. LPCs vary in their structural features, including the number and position of phenolic hydroxyl, ketone, and aliphatic groups, which results in different molecular, physicochemical, and gastrointestinal properties. In this study, we examined the impact of plant-based carrier oils (coconut, sunflower, and flaxseed oils) and LPC type (curcumin, resveratrol, and quercetin) on the in vitro gastrointestinal fate of polyphenols loaded into quillaja saponin-stabilized nanoemulsions. Coconut oil contains high levels of medium-chain saturated fatty acids (MC-SFAs), sunflower oil contains high levels of long-chain monounsaturated fatty acids (LC-MUFAs), and flaxseed oil contains high levels of long-chain polyunsaturated fatty acids (LC-PUFAs). The encapsulation efficiency and gastrointestinal stability of the LPCs were slightly lower in the MC than the LC oils. Differences in the gastrointestinal stability of the three LPCs were linked to differences in their oil-water partition coefficients. Some of the LPCs inhibited lipid digestion for certain oil types. In particular, resveratrol retarded the digestion of all three oils, but it still had the highest GIT stability and bioaccessibility. This study provides valuable information about the gastrointestinal fate of LPC-loaded nanoemulsions and highlights important differences in the behavior of LPCs with different characteristics. This knowledge may facilitate the design of more effective plant-based delivery systems for bioactive lipophilic polyphenols.

GIT stability of Hénon maps

In this paper we study the locus of generalized degree $d$ Henon maps in the parameter space $\operatorname{Rat}_d^N$ of degree $d$ rational maps $\mathbb{P}^N\to\mathbb{P}^N$ modulo the conjugation action of $\operatorname{SL}_{N+1}$. We show that Henon maps are in the GIT unstable locus if $N\ge3$ or $d\ge3$, and that they are semistable, but not stable, in the remaining case of $N=d=2$. We also give a general classification of all unstable maps in $\operatorname{Rat}_2^2$.

Relative algebro-geometric stabilities of toric manifolds

In this paper we study the relative Chow and $K$-stability of toric manifolds. First, we give a criterion for relative $K$-stability and instability of toric Fano manifolds in the toric sense. The reduction of relative Chow stability on toric manifolds will be investigated using the Hibert-Mumford criterion in two ways. One is to consider the maximal torus action and its weight polytope. We obtain a reduction by the strategy of Ono [34], which fits into the relative GIT stability detected by Székelyhidi. The other way relies on $\mathbb{C}^*$-actions and Chow weights associated to toric degenerations following Donaldson and Ross-Thomas [13, 36]. In the end, we determine the relative $K$-stability of all toric Fano threefolds and present counter-examples which are relatively $K$-stable in the toric sense but which are asymptotically relatively Chow unstable.

An Outline of the Log Minimal Model Program for the Moduli Space of Curves

ABSTRACTWe consider the log minimal model program for the moduli space of stable curves. It is widely believed now that there is a descending sequence of critical $α$ values where the log canonical model for the moduli space of stable curves with respect to $αδ$ changes, where $δ$ denotes the divisor of singular curves. We derive a conjectural formula for the critical values in two different ways: By working out the intersection theory of the moduli space of hyperelliptic curves and by computing the GIT stability of certain curves with tails and bridges. The results give a rough outline of how the log minimal model program would proceed, predicting when the log canonical model changes and which curves are to be discarded and acquired at the critical steps.

Toward GIT stability of syzygies of canonical curves

We introduce the problem of GIT stability for syzygy points of canonical curves with a view toward a GIT construction of the canonical model of the moduli space of stable curves. As the first step in this direction, we prove semi-stability of the first syzygy point for a general canonical curve of odd genus.

The homological projective dual of Sym^2 P(V)

We study the derived category of a complete intersection X of bilinear divisors in the orbifold Sym^2 P(V). Our results are in the spirit of Kuznetsov's theory of homological projective duality, and we describe a homological projective duality relation between Sym^2 P(V) and a category of modules over a sheaf of Clifford algebras on P(Sym^2 V^vee). The proof follows a recently developed strategy combining variation of GIT stability and categories of global matrix factorisations. We begin by translating D^b(X) into a derived category of factorisations on an LG model, and then apply VGIT to obtain a birational LG model. Finally, we interpret the derived factorisation category of the new LG model as a Clifford module category. In some cases we can compute this Clifford module category as the derived category of a variety. As a corollary we get a new proof of a result of Hosono and Takagi, which says that a certain pair of nonbirational Calabi-Yau 3-folds have equivalent derived categories.

A relative Hilbert–Mumford criterion

We generalize the classical Hilbert-Mumford criteria for GIT (semi-)stability in terms of one parameter subgroups of a linearly reductive group G over a field k, to the relative situation of an equivariant, projective morphism X -> Spec A to a noetherian k-algebra A. We also extend the classical projectivity result for GIT quotients: the induced morphism X^ss/G -> Spec A^G is projective. As an example of applications to moduli problems, we consider degenerations of Hilbert schemes of points.

Stability of hypersurface sections of quadric threefolds

Let $S$ be a complete intersection of a smooth quadric 3-fold $Q$ and a hypersurface of degree $d$ in ${\mathbb P}^4$. In this paper we analyze GIT stability of $S$ with respect to the natural $G=SO(5, {\mathbb C})$-action. We prove that if $d\ge 4$ and $S$ has at worst semi-log canonical singularities then $S$ is $G$-stable. Also, we prove that if $d\ge 3$ and $S$ has at worst semi-log canonical singularities then $S$ is $G$-semistable.

The GIT stability of polarized varieties via discrepancy

We prove that various GIT semistabilities of polarized varieties imply semi-log-canonicity.

COHERENT SYSTEMS AND MODULAR SUBAVRIETIES OF $\mathcal{SU}_C(r)$

Let C be an algebraic smooth complex curve of genus g > 1. The object of this paper is the study of the birational structure of certain moduli spaces of vector bundles and of coherent systems on C and the comparison of different type of notions of stability arising in moduli theory. Notably we show that in certain cases these moduli spaces are birationally equivalent to fibrations over simple projective varieties, whose fibers are GIT quotients (ℙr-1)rg// PGL (r), where r is the rank of the considered vector bundles. This allows us to compare different definitions of (semi-)stability (slope stability, α-stability, GIT stability) for vector bundles, coherent systems and point sets, and derive relations between them. In certain cases of vector bundles of low rank when C has small genus, our construction produces families of classical modular varieties contained in the Coble hypersurfaces.

Einstein metrics and GIT stability, II

Einstein metrics and GIT stability, II

GIT stability of weighted pointed curves

In the late 1970s Mumford established Chow stability of smooth unpointed genus g curves embedded by complete linear systems of degree d ≥ 2g + 1, and at about the same time Gieseker established asymptotic Hilbert stability (that is, stability of m th Hilbert points for some large values of m) under the same hypotheses. Both of them then use an indirect argument to show that nodal Deligne-Mumford stable curves are GIT stable. The case of marked points lay untouched until 2006, when Elizabeth Baldwin proved that pointed Deligne-Mumford stable curves are asymptotically Hilbert stable. (Actually, she proved this for stable maps, which includes stable curves as a special case.) Her argument is a delicate induction on g and the number of marked points n; elliptic tails are glued to the marked points one by one, ultimately relating stability of an n-pointed genus g curve to Gieseker’s result for genus g + n unpointed curves. There are three ways one might wish to improve upon Baldwin’s results. First, one might wish to construct moduli spaces of weighted pointed curves or maps; it appears that Baldwin’s proof can accommodate some, but not all, sets of weights. Second, one might wish to study Hilbert stability for small values of m; since Baldwin’s proof uses Gieseker’s proof as the base case, it is not easy to see how it could be modified to yield an approach for small m. Finally, the Minimal Model Program for moduli spaces of curves has generated interest in GIT for 2, 3, or 4-canonical linear systems; due to its use of elliptic tails, Baldwin’s proof cannot be used to study these, as elliptic tails are known to be GIT unstable in these cases. In this paper I give a direct proof that smooth curves with distinct weighted marked points are asymptotically Hilbert stable with respect to a wide range of parameter spaces and linearizations. Some of these yield the (coarse) moduli space of Deligne-Mumford stable pointed curves M g,n and Hassett’s moduli spaces of weighted pointed curves M g,A, while other linearizations may give other quotients which are birational to these and which may admit interpretations as moduli spaces. The full construction of the moduli spaces is not contained in this paper, only the proof that smooth curves with distinct weighted marked points are stable, which is the key new result needed for the construction. For this I follow Gieseker’s approach to reduce to the GIT problem to a combinatorial problem, though the solution is very different.

Einstein metrics and GIT stability

Einstein metrics and GIT stability

On K-stability of reductive varieties

G. Tian and S.K. Donaldson formulated a conjecture relating GIT stability of a polarized algebraic variety to the existence of a Kahler metric of constant scalar curvature. In [D3] Donaldson partially confirmed it in the case of projective toric varieties. In this paper we extend Donaldson’s results and computations to a new case, that of reductive varieties.