For ordinary flops, the correspondence defined by graph closure is shown to give equivalence of Chow motives and preserve Poincaré pairing.In case simple this preserves big quantum cohomology ring after an analytic continuation over extended Kähler moduli space.For Mukai it that birational map for local models deformation equivalent isomorphisms.This implies induces isomorphisms on full theory all corrections attached extremal ray vanish.

Hydrochloric acid (HCl) and caustic (NaOH) are among the most widely used chemicals by water industry. Direct anodic electrochemical HCl production electrolysis has not been successful as current commercially available electrodes prone to chlorine formation. This study presents an innovative technology simultaneously generating NaOH from NaCl using a Mn0.84Mo0.16O2.23 oxygen evolution electrode during electrolysis. The results showed that protons could be anodically generated at high...

In this work, we continue our study initiated in [11]. We show that the generating functions of Gromov–Witten invariants with ancestors are invariant under a simple flop, for all genera, after an analytic continuation extended Kähler moduli space.

Recently, naturally occurring magnetite (Fe3O4) has emerged as a new material for sulfide control in sewers. However, unrefined could have high heavy metal contents (e.g., Cr, Zn, Ni, Sn, etc.) and the capacity to remove dissolved is reasonably limited due relatively large particle sizes. To overcome drawbacks of we used an electrochemical system with mild steel sacrificial electrodes in-situ generate strength solutions plate-like nanoparticles (MNP). MNP size range between 120 160 nm were...

In this paper we prove the invariance of quantum rings for general ordinary flops, whose local models are certain non-split toric bundles over arbitrary smooth bases.An essential ingredient in proof is a splitting principle which reduces statement Gromov-Witten theory on to case split bundles.

This is the second of a sequence papers proving quantum invariance for ordinary flops over an arbitrary smooth base.In this paper, we complete proof big rings under split type.To achieve that, several new ingredients are introduced.One Leray-Hirsch theorem local model (a certain toric bundle) which extends D-module Dubrovin connection on base by Picard-Fuchs system fibers.Non-split as well further applications will be discussed in subsequent papers.In particular, splitting principle...

Caustic shock-loading and oxygen injection are commonly used by the water industry for biofilm sulfide control in sewers. can be produced onsite from wastewater using a two-compartment electrochemical cell. This avoids need import storage of caustic soda, which typically represents cost hazard. An issue limiting practical implementation this approach is occurrence membrane scaling due to almost universal presence Ca2+ Mg2+ wastewater. It results rapid increase cell voltage, thereby...

The paper is Part III of our ongoing project to study a case Crepant Transformation Conjecture: K-equivalence Conjecture for ordinary flops. In this we prove the invariance quantum rings general flops, whose local models are certain non-split toric bundles over arbitrary smooth base. An essential ingredient in proof splitting principle, which reduces statement Gromov--Witten theory on split bundles.

Introduction.Let X be an (n + 1)-dimensional smooth complex projective variety and let D a ample divisor of with inclusion map i : -> X.The well-known Weak Lefschetz Theorem (see [GrHa]) asserts that the restriction z* H k (X\ Z) ->► (D\ is isomorphism for < n-1 compatible Hodge decomposition.For n > 3 one deduces from these results Pic(X) Pic(D) also isomorphism.Grothendieck has shown this statement true over any algebraically closed field [Hart].While line bundle on always restricts to D,...

This is an expanded version of the third author's lecture in String-Math 2015 at Sanya. It summarizes some our works quantum cohomology. After reviewing Lefschetz and Leray--Hirsch, we discuss their applications to functoriality properties under special smooth flops, flips blow-ups. Finally, for conifold transitions Calabi--Yau 3-folds, formulations small resolutions (blow-ups along Weil divisors) are sketched.

This is the first of a sequence papers proving quantum invariance under ordinary flops over an arbitrary smooth base.In this part, we determine defect cup product canonical correspondence and show that it corrected by small attached to extremal ray.We then perform various reductions reduce problem local models.In part II ("Invariance rings II", Algebraic Geometry, 2016), develop Leray-Hirsch theorem use big cohomology ring invariant analytic continuation in Kähler moduli space for split...

In this article, we find some diagonal hypersurfaces that admit crepant resolutions. We also give a criterion for unique factorization domains.

For projective conifold transitions between Calabi-Yau threefolds $X$ and $Y$, with close to $Y$ in the moduli, we show that combined information provided by $A$ model (Gromov--Witten theory all genera) $B$ (variation of Hodge structures) on $X$, linked along vanishing cycles, determines corresponding $Y$. Similar result holds reverse direction when exceptional curves.

For projective conifold transitions between Calabi–Yau threefolds $X$ and $Y$, with close to $Y$ in the moduli, we show that combined information provided by $A$ model (Gromov–Witten theory all genera) $B$ (variation of Hodge structures) on $X$, linked along vanishing cycles, determines corresponding $Y$. Similar result holds reverse direction when exceptional curves.

This is the second of a sequence papers proving quantum invariance for ordinary flops over an arbitrary smooth base. In this paper, we complete proof big rings under splitting type. To achieve that, several new ingredients are introduced. One Leray--Hirsch theorem local model (a certain toric bundle) which extends D module Dubrovin connection on base by Picard--Fuchs system fibers. Nonsplit as well further applications will be discussed in subsequent papers. particular, principle developed...

Let $\Sigma$ and $\Sigma'$ be two refinements of a fan $\Sigma_0$ $f \colon X_{\Sigma} \dashrightarrow X_{\Sigma'}$ the birational map induced by $X_{\Sigma} \rightarrow X_{\Sigma_0} \leftarrow X_{\Sigma'}$. We show that graph closure $\overline{\Gamma}_f$ is not necessarily normal toric variety we give combinatorial criterion for its normality. In contrast to it, $f$ being flop/flip, scheme-theoretic fiber product $X:=X_{\Sigma}\mathop{\times}\limits_{X_{\Sigma_0}}X_{\Sigma'}$ in general...

<!-- *** Custom HTML --> In the joint paper [8] with Y.-P. Lee and C.-L. Wang, we have shown that big quantum ring is invariant under $\mathbb{P}^r$ flops of splitting type, after an analytic continuation over extended Kähler moduli space. It a generalization our previous work for case simple [7]. this note, I would like to outline results concentrate mainly on detailed study called $\mathbb{P}^1$ type $(k + 2, k)$.