Cluster States for Continuous-Variable Multipartite Entanglement

We introduce a new class of continuous-variable (CV) multipartite entangled states, the CV cluster states, which might be generated from squeezing and kerr-like interaction. The entanglement properties of these states are studied in terms of classical communication and local operations. The quantum teleportation network with cluster states is investigated. The graph states as the general forms of cluster states are presented, which may be used to generate CV Greenberger-Horne-Zeilinger states by simply local measurements and classical communication. A chain for one-dimensional example of cluster states can be readily experimentally produced only with squeezed light and beamsplitters.Comment: 4 page

Hodge Cohomology Criteria For Affine Varieties

We give several new criteria for a quasi-projective variety to be affine. In particular, we prove that an algebraic manifold $Y$ with dimension $n$ is affine if and only if $H^i(Y, \Omega^j_Y)=0$ for all $j\geq 0$, $i>0$ and $\kappa(D, X)=n$, i.e., there are $n$ algebraically independent nonconstant regular functions on $Y$, where $X$ is the smooth completion of $Y$, $D$ is the effective boundary divisor with support $X-Y$ and $\Omega^j_Y$ is the sheaf of regular $j$-forms on $Y$. This proves Mohan Kumar's affineness conjecture for algebraic manifolds and gives a partial answer to J.-P. Serre's Steinness question \cite{36} in algebraic case since the associated analytic space of an affine variety is Stein [15, Chapter VI, Proposition 3.1].Comment: 19 page

Threefolds with Vanishing Hodge Cohomology

We consider algebraic manifolds $Y$ of dimension 3 over $\Bbb{C}$ with $H^i(Y, \Omega^j_Y)=0$ for all $j\geq 0$ and $i>0$. Let $X$ be a smooth completion of $Y$ with $D=X-Y$, an effective divisor on $X$ with normal crossings. If the $D$-dimension of $X$ is not zero, then $Y$ is a fibre space over a smooth affine curve $C$ (i.e., we have a surjective morphism from $Y$ to $C$ such that general fibre is smooth and irreducible) such that every fibre satisfies the same vanishing condition. If an irreducible smooth fibre is not affine, then the Kodaira dimension of $X$ is $-\infty$ and the $D$-dimension of X is 1. We also discuss sufficient conditions from the behavior of fibres or higher direct images to guarantee the global vanishing of Hodge cohomology and the affineness of $Y$.Comment: 24 pages, accepted by Transactions of AM

Bertini Type Theorems

Let $X$ be a smooth irreducible projective variety of dimension at least 2 over an algebraically closed field of characteristic 0 in the projective space ${\mathbb{P}}^n$. Bertini's Theorem states that a general hyperplane $H$ intersects $X$ with an irreducible smooth subvariety of $X$. However, the precise location of the smooth hyperplane section is not known. We show that for any $q\leq n+1$ closed points in general position and any degree $a>1$, a general hypersurface $H$ of degree $a$ passing through these $q$ points intersects $X$ with an irreducible smooth codimension 1 subvariety on $X$. We also consider linear system of ample divisors and give precise location of smooth elements in the system. Similar result can be obtained for compact complex manifolds with holomorphic maps into projective spaces.Comment: 20 page

Monochromatic Sumset Without the use of large cardinals

We show in this note that in the forcing extension by $Add(\omega,\beth_{\omega})$, the following Ramsey property holds: for any $r\in \omega$ and any $f: \mathbb{R}\to r$, there exists an infinite $X\subset \mathbb{R}$ such that $X+X$ is monochromatic under $f$. We also show the Ramsey statement above is true in $\mathrm{ZFC}$ when $r=2$. This answers two questions by Komj\'ath, Leader, Russell, Shelah, Soukup and Vidny\'anszky

A Literature Survey of Cooperative Caching in Content Distribution Networks

Content distribution networks (CDNs) which serve to deliver web objects (e.g., documents, applications, music and video, etc.) have seen tremendous growth since its emergence. To minimize the retrieving delay experienced by a user with a request for a web object, caching strategies are often applied - contents are replicated at edges of the network which is closer to the user such that the network distance between the user and the object is reduced. In this literature survey, evolution of caching is studied. A recent research paper [15] in the field of large-scale caching for CDN was chosen to be the anchor paper which serves as a guide to the topic. Research studies after and relevant to the anchor paper are also analyzed to better evaluate the statements and results of the anchor paper and more importantly, to obtain an unbiased view of the large scale collaborate caching systems as a whole.Comment: 5 pages, 5 figure

On the DD-dimension of a certain type of threefolds

Let $Y$ be an algebraic manifold of dimension 3 with $H^i(Y, \Omega^j_Y)=0$ for all $j\geq 0$, $i>0$ and $h^0(Y, {\mathcal{O}}_Y) > 1$. Let $X$ be a smooth completion of $Y$ such that the boundary $X-Y$ is the support of an effective divisor $D$ on $X$ with simple normal crossings. We prove that the $D$-dimension of $X$ cannot be 2, i.e., either any two nonconstant regular functions are algebraically dependent or there are three algebraically independent nonconstant regular functions on $Y$. Secondly, if the $D$-dimension of $X$ is greater than 1, then the associated scheme of $Y$ is isomorphic to Spec$\Gamma(Y, {\mathcal{O}}_Y)$. Furthermore, we prove that an algebraic manifold $Y$ of any dimension $d\geq 1$ is affine if and only if $H^i(Y, \Omega^j_Y)=0$ for all $j\geq 0$, $i>0$ and it is regularly separable, i.e., for any two distinct points $y_1$, $y_2$ on $Y$, there is a regular function $f$ on $Y$ such that $f(y_1)\neq f(y_2)$.Comment: 14 page

A tail cone version of the Halpern-L\"auchli theorem at a large cardinal

The classical Halpern-L\"auchli theorem states that for any finite coloring of a finite product of finitely branching perfect trees of height $\omega$, there exist strong subtrees sharing the same level set such that tuples consisting of elements lying on the same level get the same color. Relative to large cardinals, we establish the consistency of a tail cone version of the Halpern-L\"auchli theorem at large cardinal, which, roughly speaking, deals with many colorings simultaneously and diagonally. Among other applications, we generalize a polarized partition relation on rational numbers due to Laver and Galvin to one on linear orders of larger saturation.Comment: Updated versio

Rado's conjecture and its Baire version

Rado's Conjecture is a compactness/reflection principle that says any nonspecial tree of height $\omega_1$ has a nonspecial subtree of size $\leq \aleph_1$. Though incompatible with Martin's Axiom, Rado's Conjecture turns out to have many interesting consequences that are consequences of forcing axioms. In this paper, we obtain consistency results concerning Rado's Conjecture and its Baire version. In particular, we show a fragment of PFA, that is the forcing axiom for \emph{Baire Indestructibly proper forcings}, is compatible with the Baire Rado's Conjecture. As a corollary, Baire Rado's Conjecture does not imply Rado's Conjecture. Then we discuss the strength and limitations of the Baire Rado's Conjecture regarding its interaction with simultaneous stationary reflection and some families of weak square principles. Finally we investigate the influence of the Rado's Conjecture on some polarized partition relations.Comment: Incorporated comments and corrections from the refere

Review of Around the Texts of Writing Center Work: An Inquiry-Based Approach to Tutor Education, by R. Mark Hall

University Writing Cente