Fourfolds

We have found a "non-purely-constructive" method of acquiring algebraic cycles involving multiple steps. This note tries to present the main idea in the last step by concentrating on an example of 4-folds. The method demonstrates a contrast to traditional constructions

The cone construction via intersection theory

We show a method in constructing algebraic cycles via intersection theory. It leads to a proof of the Lefschetz standard conjecture.Comment: arXiv admin note: text overlap with arXiv:1801.0532

Cone construction II

This is the second part of two parts, titled " cone construction". In this part we prove the Lefschetz cohomologicity of the cone operator $Con$.Comment: It is combined with the revision of cone construction

Equivalence of Coniveaus

On a smooth projective variety over the complex numbers, there is the coniveau from the coniveau filtration, which is called geometric coniveau. On the same variety, there is another coniveau from the maximal sub-Hodge structure, which is called Hodge coniveau. In this paper we show they are equivalent

Real intersection theory (I)

We introduce the notion of Lebesgue currents. They are a special type of currents involving Lebesgue measure. We apply it to define the intersection of singular cycles, which provides the foundation to the real intersection theory.Comment: arXiv admin note: text overlap with arXiv:1801.0136

Leveled sub-cohomology

In this paper we define a functor-- leveled sub-cohomology. (It bears no relation with the level of elliptic curves). It is based on leveled cycles on a smooth projective variety, and will be expected to reveal a structure in the level

Threefolds

This is an example on the cohomology of threefolds

Rational curves on complete intersection Calabi-Yau 3-folds

We prove the following results. If $X_3$ is a generic complete intersection Calabi-Yau 3-fold, (1) then for each natural number $d$ there exists a rational map \par\hspace{1 cc} $c\in Hom_{bir}(\mathbf P^1, X_3)$ of $deg(c(\mathbf P^1))=d$, (2) further more all such $c$ are immersions satisfying \begin{equation} N_{c(\mathbf P^1)/ X_3}\simeq \mathcal O_{\mathbf P^1}(-1)\oplus \mathcal O_{\mathbf P^1}(-1)

A remark on the local density approximation with the gradient corrections and the Xα_\alpha method

We report that the solids with narrow valence bands cannot be described by the local density approximation with the gradient corrections in the density functional theory as well as the X$_\alpha$ method. In particular, in the case of completely filled valence bands, the work function is significantly underestimated by these methods for such types of solids. Also, we figured out that these deficiencies cannot be cured by the self-interaction-corrected-local-density-approximation method.Comment: 6 pages, 0 figure

Classical approach to the graph isomorphism problem using quantum walks

Given the extensive application of classical random walks to classical algorithms in a variety of fields, their quantum analogue in quantum walks is expected to provide a fruitful source of quantum algorithms. So far, however, such algorithms have been scarce. In this work, we enumerate some important differences between quantum and classical walks, leading to their markedly different properties. We show that for many practical purposes, the implementation of quantum walks can be efficiently achieved using a classical computer. We then develop both classical and quantum graph isomorphism algorithms based on discrete-time quantum walks. We show that they are effective in identifying isomorphism classes of large databases of graphs, in particular groups of strongly regular graphs. We consider this approach to represent a promising candidate for an efficient solution to the graph isomorphism problem, and believe that similar methods employing quantum walks, or derivatives of these walks, may prove beneficial in constructing other algorithms for a variety of purposes