180 research outputs found
Interlocking structure design and assembly
Many objects in our life are not manufactured as whole rigid pieces. Instead, smaller components are made to be later assembled into larger structures. Chairs are assembled from wooden pieces, cabins are made of logs, and buildings are constructed from bricks. These components are commonly designed by many iterations of human thinking. In this report, we will look at a few problems related to interlocking components design and assembly. Given an atomic object, how can we design a package that holds the object firmly without a gap in-between? How many pieces should the package be partitioned into? How can we assemble/extract each piece? We will attack this problem by first looking at the lower bound on the number of pieces, then at the upper bound. Afterwards, we will propose a practical algorithm for designing these packages. We also explore a special kind of interlocking structure which has only one or a small number of movable pieces. For example, a burr puzzle. We will design a few blocks with joints whose combination can be assembled into almost any voxelized 3D model. Our blocks require very simple motions to be assembled, enabling robotic assembly. As proof of concept, we also develop a robot system to assemble the blocks. In some extreme conditions where construction components are small, controlling each component individually is impossible. We will discuss an option using global controls. These global controls can be from gravity or magnetic fields. We show that in some special cases where the small units form a rectangular matrix, rearrangement can be done in a small space following a technique similar to bubble sort algorithm
On Sunflowers and Matrix Multiplication
We present several variants of the sunflower conjecture of ErdÅs and Rado and discuss the relations among them.
We then show that two of these conjectures (if true) imply negative answers to questions of Coppersmith and Winograd and Cohn et al. regarding possible approaches for obtaining fast matrix multiplication algorithms. Specifically, we show that the ErdÅs-Rado sunflower conjecture (if true) implies a negative answer to the āno three disjoint equivoluminous subsetsā question of Coppersmith and Winograd; we also formulate a āmulticoloredā sunflower conjecture in Znā and show that (if true) it implies a negative answer to the āstrong USPā conjecture of Cohn et al. (although it does not seem to impact a second conjecture in that paper or the viability of the general group theoretic approach). A surprising consequence of our results is that the Coppersmith-Winograd conjecture actually implies the Cohn et al. conjecture.
The multicolored sunflower conjecture in Znā is a strengthening of the well-known (ordinary) sunflower conjecture in Znā, and we show via our connection that a construction of Cohn et al. yields a lower bound of (2.51...)^n on the size of the largest multicolored 3-sunflower-free set, which beats the current best known lower bound of (2.21...)^n on the size of the largest 3-sunflower-free set in Znā
Existence of r-fold perfect (v,K,1)-Mendelsohn designs with Kā{4,5,6,7}
AbstractLet v be a positive integer and let K be a set of positive integers. A (v,K,1)-Mendelsohn design, which we denote briefly by (v,K,1)-MD, is a pair (X,B) where X is a v-set (of points) and B is a collection of cyclically ordered subsets of X (called blocks) with sizes in the set K such that every ordered pair of points of X are consecutive in exactly one block of B. If for all t=1,2,ā¦,r, every ordered pair of points of X are t-apart in exactly one block of B, then the (v,K,1)-MD is called an r-fold perfect design and denoted briefly by an r-fold perfect (v,K,1)-MD. If K={k} and r=kā1, then an r-fold perfect (v,{k},1)-MD is essentially the more familiar (v,k,1)-perfect Mendelsohn design, which is briefly denoted by (v,k,1)-PMD. In this paper, we investigate the existence of r-fold perfect (v,K,1)-Mendelsohn designs for a specified set K which is a subset of {4, 5, 6, 7} containing precisely two elements
Disjoint skolem-type sequences and applications
Let D = {iā, iā,..., in} be a set of n positive integers. A Skolem-type sequence
of order n is a sequence of i such that every i ā D appears exactly twice in the
sequence at position aįµ¢ and bįµ¢, and |bįµ¢ - aįµ¢| = i. These sequences might contain
empty positions, which are filled with 0 elements and called hooks. For example,
(2; 4; 2; 0; 3; 4; 0; 3) is a Skolem-type sequence of order n = 3, D = f2; 3; 4g and two
hooks. If D = f1; 2; 3; 4g we have (1; 1; 4; 2; 3; 2; 4; 3), which is a Skolem-type sequence
of order 4 and zero hooks, or a Skolem sequence.
In this thesis we introduce additional disjoint Skolem-type sequences of order n
such as disjoint (hooked) near-Skolem sequences and (hooked) Langford sequences.
We present several tables of constructions that are disjoint with known constructions
and prove that our constructions yield Skolem-type sequences. We also discuss the
necessity and sufficiency for the existence of Skolem-type sequences of order n where
n is positive integers
Approximate unitary -designs by short random quantum circuits using nearest-neighbor and long-range gates
We prove that -depth local random quantum circuits
with two qudit nearest-neighbor gates on a -dimensional lattice with n
qudits are approximate -designs in various measures. These include the
"monomial" measure, meaning that the monomials of a random circuit from this
family have expectation close to the value that would result from the Haar
measure. Previously, the best bound was due to
Brandao-Harrow-Horodecki (BHH) for . We also improve the "scrambling" and
"decoupling" bounds for spatially local random circuits due to Brown and Fawzi.
One consequence of our result is that assuming the polynomial hierarchy (PH)
is infinite and that certain counting problems are -hard on average,
sampling within total variation distance from these circuits is hard for
classical computers. Previously, exact sampling from the outputs of even
constant-depth quantum circuits was known to be hard for classical computers
under the assumption that PH is infinite. However, to show the hardness of
approximate sampling using this strategy requires that the quantum circuits
have a property called "anti-concentration", meaning roughly that the output
has near-maximal entropy. Unitary 2-designs have the desired anti-concentration
property. Thus our result improves the required depth for this level of
anti-concentration from linear depth to a sub-linear value, depending on the
geometry of the interactions. This is relevant to a recent proposal by the
Google Quantum AI group to perform such a sampling task with 49 qubits on a
two-dimensional lattice and confirms their conjecture that depth
suffices for anti-concentration. We also prove that anti-concentration is
possible in depth O(log(n) loglog(n)) using a different model
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