5,973 research outputs found
Cryptography from tensor problems
We describe a new proposal for a trap-door one-way function. The new proposal belongs to the "multivariate quadratic" family but the trap-door is different from existing methods, and is simpler
Topological Optimization of the Evaluation of Finite Element Matrices
We present a topological framework for finding low-flop algorithms for
evaluating element stiffness matrices associated with multilinear forms for
finite element methods posed over straight-sided affine domains. This framework
relies on phrasing the computation on each element as the contraction of each
collection of reference element tensors with an element-specific geometric
tensor. We then present a new concept of complexity-reducing relations that
serve as distance relations between these reference element tensors. This
notion sets up a graph-theoretic context in which we may find an optimized
algorithm by computing a minimum spanning tree. We present experimental results
for some common multilinear forms showing significant reductions in operation
count and also discuss some efficient algorithms for building the graph we use
for the optimization
Conserved- and zero-mean quadratic quantities in oscillatory systems
We study quadratic functionals of the variables of a linear oscillatory system and their derivatives. We show that such functionals are partitioned in conserved quantities and in trivially- and intrinsic zero-mean quantities. We also state an equipartition of energy principle for oscillatory systems
Identification and data-driven model reduction of state-space representations of lossless and dissipative systems from noise-free data
We illustrate procedures to identify a state-space representation of a lossless- or dissipative system from a given noise-free trajectory; important special cases are passive- and bounded-real systems. Computing a rank-revealing factorization of a Gramian-like matrix constructed from the data, a state sequence can be obtained; state-space equations are then computed solving a system of linear equations. This idea is also applied to perform model reduction by obtaining a balanced realization directly from data and truncating it to obtain a reduced-order mode
An excursion from enumerative goemetry to solving systems of polynomial equations with Macaulay 2
Solving a system of polynomial equations is a ubiquitous problem in the
applications of mathematics. Until recently, it has been hopeless to find
explicit solutions to such systems, and mathematics has instead developed deep
and powerful theories about the solutions to polynomial equations. Enumerative
Geometry is concerned with counting the number of solutions when the
polynomials come from a geometric situation and Intersection Theory gives
methods to accomplish the enumeration.
We use Macaulay 2 to investigate some problems from enumerative geometry,
illustrating some applications of symbolic computation to this important
problem of solving systems of polynomial equations. Besides enumerating
solutions to the resulting polynomial systems, which include overdetermined,
deficient, and improper systems, we address the important question of real
solutions to these geometric problems.
The text contains evaluated Macaulay 2 code to illuminate the discussion.
This is a chapter in the forthcoming book "Computations in Algebraic Geometry
with Macaulay 2", edited by D. Eisenbud, D. Grayson, M. Stillman, and B.
Sturmfels. While this chapter is largely expository, the results in the last
section concerning lines tangent to quadrics are new.Comment: LaTeX 2e, 22 pages, 1 .eps figure. Source file (.tar.gz) includes
Macaulay 2 code in article, as well as Macaulay 2 package realroots.m2
Macaulay 2 available at http://www.math.uiuc.edu/Macaulay2 Revised with
improved exposition, references updated, Macaulay 2 code rewritten and
commente
On the Exponentials of Some Structured Matrices
In this note explicit algorithms for calculating the exponentials of
important structured 4 x 4 matrices are provided. These lead to closed form
formulae for these exponentials. The techniques rely on one particular Clifford
Algebra isomorphism and basic Lie theory. When used in conjunction with
structure preserving similarities, such as Givens rotations, these techniques
extend to dimensions bigger than four.Comment: 19 page
Supertropical Quadratic Forms I
We initiate the theory of a quadratic form over a semiring . As
customary, one can write where is a
companion bilinear form. But in contrast to the ring-theoretic case, the
companion bilinear form need not be uniquely defined. Nevertheless, can
always be written as a sum of quadratic forms where
is quasilinear in the sense that and is rigid in the sense that it has a unique companion. In
case that is a supersemifield (cf. Definition 4.1 below) and is defined
on a free -module, we obtain an explicit classification of these
decompositions and of all companions of .
As an application to tropical geometry, given a quadratic form
on a free module over a commutative ring and a supervaluation
with values in a supertropical semiring [5], we define -
after choosing a base of - a quadratic form on the free module over the semiring . The analysis
of quadratic forms over a supertropical semiring enables one to measure the
"position" of with respect to via .Comment: 31 page
Testing isomorphism of graded algebras
We present a new algorithm to decide isomorphism between finite graded
algebras. For a broad class of nilpotent Lie algebras, we demonstrate that it
runs in time polynomial in the order of the input algebras. We introduce
heuristics that often dramatically improve the performance of the algorithm and
report on an implementation in Magma
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