351,165 research outputs found

### Surgery on links with unknotted components and three-manifolds

It is shown that any closed three-manifold M obtained by integral surgery on
a knot in the three-sphere can always be constructed from integral surgeries on
a 3-component link L with each component being an unknot in the three-sphere.
It is also interesting to notice that infinitely many different integral
surgeries on the same link L could give the same three-manifold M.Comment: 10 pages, 8 figure

### What is NP? - Interpretation of a Chinese paradox "white horse is not horse"

The notion of nondeterminism has disappeared from the current definition of
NP, which has led to ambiguities in understanding NP, and caused fundamental
difficulties in studying the relation P versus NP. In this paper, we question
the equivalence of the two definitions of NP, the one defining NP as the class
of problems solvable by a nondeterministic Turing machine in polynomial time,
and the other defining NP as the class of problems verifiable by a
deterministic Turing machine in polynomial time, and reveal cognitive biases in
this equivalence. Inspired from a famous Chinese paradox white horse is not
horse, we further analyze these cognitive biases. The work shows that these
cognitive biases arise from the confusion between different levels of
nondeterminism and determinism, due to the lack of understanding about the
essence of nondeterminism. Therefore, we argue that fundamental difficulties in
understanding P versus NP lie firstly at cognition level, then logic level

### On free $Z_p$-torus actions in dimension two and three

We confirm the Halperin-Carlsson Conjecture for free $Z_p$-torus actions (p
is a prime) on 2-dimensional finite CW-complexes and free $Z_2$-torus actions
on compact 3-manifolds.Comment: 26 pages, no figure. The contents of the paper are reorganized and
some proofs are simplifie

### On lower bounds of the sum of multigraded Betti numbers of simplicial complexes

We find some general lower bounds of the sum of certain families of
multigraded Betti numbers of any simplicial complex with a vertex coloring.Comment: 15 pages, 2 figures. Minor revisions are made (two pictures and some
new references are added

### On the constructions of free and locally standard Z_2-torus actions on Manifolds

We introduce an elementary way of constructing principal (Z_2)^m-bundles over
compact smooth manifolds. In addition, we will define a general notion of
locally standard (Z_2)^m-actions on closed manifolds for all m>0, and then give
a general way to construct all such (Z_2)^m-actions from the orbit space. Some
related topology problems are also studied.Comment: 28 pages, 12 figures, some minor revisions are made, one picture and
one reference are added

### Quantum Boson Algebra and Poisson Geometry of the Flag Variety

In his work on crystal bases \cite{Kas}, Kashiwara introduced a certain
degeneration of the quantized universal enveloping algebra of a semi-simple Lie
algebra $\mathfrak g$, which he called a quantum boson algebra. In this paper,
we construct Kashiwara operators associated to all positive roots and use them
to define a variant of Kashiwara's quantum boson algebra. We show that a
quasi-classical limit of the positive half of our variant is a Poisson algebra
of the form $(P \simeq \mathbb C[\mathfrak n^{\ast}], \, \{~~,~~\}_P)$, where
$\mathfrak n$ is the positive part of $\mathfrak g$ and $\{~~,~~\}_P$ is a
Poisson bracket that has the same rank as, but is different from, the
Kirillov-Kostant bracket $\{~~,~~\}_{KK}$ on $\mathfrak n^{\ast}$. Furthermore,
we prove that, in the special case of type $A$, any linear combination $a_1
\{~~,~~\}_P + a_2 \{~~,~~\}_{KK}$, $a_1, a_2 \in \mathbb C$, is again a Poisson
bracket. In the general case, we establish an isomorphism of $P$ and the
Poisson algebra of regular functions on the open Bruhat cell in the flag
variety. In type $A$, we also construct a Casimir function on the open Bruhat
cell, together with its quantization, which may be thought of as an analog of
the linear function on $\mathfrak n^{\ast}$ defined by a root vector for the
highest root

### Cubes and Generalized Real Bott Manifolds

We define a notion of facets-pairing structure and its seal space on a nice
manifold with corners. We will study facets-pairing structures on any cube in
detail and investigate when the seal space of a facets-pairing structure on a
cube is a closed manifold. In particular, for any binary square matrix $A$ with
zero diagonal in dimension n, there is a canonical facets-pairing structure
$F_A$ on the n-dimensional cube. We will show that all the closed manifolds
that we can obtain from the seal spaces of such $F_A$'s are neither more nor
less than all the generalized real Bott manifolds --- a special class of real
toric manifolds introduced by Choi, Masuda and Suh.Comment: Some small changes were made to the previous version. The
introduction part was expanded and a new reference was adde

### Gaiotto's Lagrangian subvarieties via loop groups

The purpose of this note is to give a simple proof of the fact that a certain
substack, defined in [2], of the moduli stack $T^{\ast}Bun_G(\Sigma)$ of Higgs
bundles over a curve $\Sigma$, for a connected, simply connected semisimple
group $G$, possesses a Lagrangian structure. The substack, roughly speaking,
consists of images under the moment map of global sections of principal
$G$-bundles over $\Sigma$ twisted by a smooth symplectic variety with a
Hamiltonian $G$-action

### A quantum homomorphic encryption scheme for polynomial-sized circuits

Quantum homomorphic encryption (QHE) is an encryption method that allows
quantum computation to be performed on one party's private data with the
program provided by another party, without revealing much information about the
data nor about the program to the opposite party. It is known that
information-theoretically-secure QHE for circuits of unrestricted size would
require exponential resources, and efficient computationally-secure QHE schemes
for polynomial-sized quantum circuits have been constructed. In this paper we
first propose a QHE scheme for a type of circuits of polynomial depth, based on
the rebit quantum computation formalism. The scheme keeps the restricted type
of data perfectly secure. We then propose a QHE scheme for a larger class of
polynomial-depth quantum circuits, which has partial data privacy. Both schemes
have good circuit privacy. We also propose an interactive QHE scheme with
asymptotic data privacy, however, the circuit privacy is not good, in the sense
that the party who provides the data could cheat and learn about the circuit.
We show that such cheating would generally affect the correctness of the
evaluation or cause deviation from the protocol. Hence the cheating can be
caught by the opposite party in an interactive scheme with embedded
verifications. Such scheme with verification has a minor drawback in data
privacy. Finally, we show some methods which achieve some nontrivial level of
data privacy and circuit privacy without resorting to allowing early
terminations, in both the QHE problem and in secure evaluation of classical
functions. The entanglement and classical communication costs in these schemes
are polynomial in the circuit size and the security parameter (if any).Comment: 29 pages, 2 figures. Revised Section VIII, among other minor fixe

### Small Cover and Halperin-Carlsson Conjecture

We prove that the Halperin-Carlsson conjecture holds for any free (Z_2)^m
action on a compact manifold whose orbit space is a small cover. In addition,
we show that if the total space of a principal (Z_2)^m bundle over a small
cover is connected, it must be equivalent to a partial quotient of the
corresponding real moment-angle manifold with some canonical Z_2-torus action.Comment: 20 pages, 4 figure. The article is significantly expanded from the
previous version, and some references are adde

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