1,389 research outputs found
Multiplying matrices using n arithmetic operations
It is widely known that the lower bound for the algorithmic complexity of
square matrix multiplication resorts to at least arithmetic operations.
The justification builds upon the following reasoning: given that there are numbers in the input matrices, any algorithm necessarily must operate on
each at least once. In this paper, we show that this is not necessarily the
case for certain instances of the problem, for instance matrices with natural
number entries. We present an algorithm performing a single multiplication and
sums, therefore using n arithmetic operations. The ingenuity of the
approach relies on encoding the original elements as two numbers of much
greater magnitude. Thus, though processing each of the inputs at least once, it
relies on a lower count of arithmetic operations. In the computational model
used to analyze this problem, such encoding operation is not available, thus it
is not clear this work affects the currently accepted complexity results for
matrix multiplication, but the new algorithm complexity (when taking into
account the encodings) is operations. In addition, given the
exponential increase in multiplication operands magnitude, its practical usage
is constrained to certain instances of the problem. Nonetheless, this work
presents a novel mathematically inspired algorithm while pointing towards an
alternative research path, which opens the possibility of novel algorithms and
a taxonomy of matrix multiplications and associated complexities
Life history of the coastal tailed frog (Ascaphus truei) across an elevational gradient
The life history of a species is described in terms of its growth, longevity, and reproduction. Unsurprisingly, life history traits are known to vary in many taxa across environmental gradients. In the case of amphibians, species at high elevations and latitudes tend to have shorter breeding seasons, shorter activity periods, longer larval periods, reach sexual maturity at older ages, and produce fewer and larger clutches per year.
The Coastal Tailed Frog (Ascaphus truei) is an ideal species for the study of geographic variation in life history because it ranges across most of the Pacific Northwest from northern California into British Columbia, and along its range it varies geographically in larval period and morphology. During a California Department of Fish and Wildlife restoration project in the Trinity Alps Wilderness, I had incidental captures of Coastal Tailed Frog larvae and adults. To date, no population across the species’ range has been described above 2000m. These populations in the Trinity Alps range from 150m to over 2100m in elevation, and those that are in the higher part of the range are likely living at the species’ maximum elevational limit.
In this study, I examined size, growth, larval period, size at sexual maturity, and longevity of A. truei across populations along an elevational gradient in the Klamath Mountains of northern California. I calculated growth rates and movement by individually marking tadpoles and post-metamorphic frogs with visual implant elastomer (VIE), then tracking them from May through October of 2018. I described the length of the larval period using length-density histograms to visualize larval cohorts, I determined size at sexual maturity using secondary sexual characteristics of post-metamorphic frogs, and I determined longevity using skeletochronology.
I found that the larval period of A. truei in the Klamath Mountains of northern California ranges from two years in low and mid-elevations, to at least three years in high elevations. I also found decreased body size and increased growth rates of tadpoles with increasing elevation. Post-metamorphic frogs grew at similar rates as previously described coastal California populations. There was high site fidelity and significantly greater movement during the months of June and August in post-metamorphic animals. Frogs in the high elevations are capable of great longevity, with a maximum observed age estimated at eight years post-metamorphosis.
The high elevation populations described here have the longest larval period documented in California. This study also provides the first field estimates of larval growth rates and the first longevity estimates of post metamorphic frogs in California. Future laboratory experiments will be necessary to separate phenotypic plasticity of life history traits from true genetic differences between A. truei populations in the Klamath Mountains of northern California, as potential explanations for the variation seen
Unraveling Functional Equations in Composition Algebra: Resolving Conjectures and Examining Implications
We address the conjectures left by the recent article by Ferreira et al.
titled ``Commuting maps and identities with inverses on alternative division
rings.'' We also present an example showing the necessity of the conditions of
the results that answer the conjectures.Comment: 1
Exploring Functional Identities: From Division Rings to Matrix Algebras
In this paper, we tackle unresolved inquiries by Ferreira et al. \cite{bruno}
in their recent publication, ``Functional Identity on Division Algebras". We
delve into the intricate behavior of additive functions on matrix algebras over
division rings through rigorous analysis and theorem-proving. Our findings
offer valuable insights into the nature of these functions and their
implications for algebraic structures.Comment: 1
Matrices as arrows! A biproduct approach to typed linear algebra
Motivated by the need to formalize generation of fast running code for linear algebra applications, we show how an index-free, calculational approach to matrix algebra can be developed by regarding matrices as morphisms of a category with biproducts. This shifts the traditional view of matrices as indexed structures to a type-level perspective analogous to that of the pointfree algebra of programming. The derivation of fusion, cancellation and abide laws from the biproduct equations makes it easy to calculate algorithms implementing matrix multiplication, the kernel operation of matrix algebra, ranging from its divide-and-conquer version to the conventional, iterative one.
From errant attempts to learn how particular products and coproducts emerge from biproducts, we not only rediscovered block-wise matrix com- binators but also found a way of addressing other operations calculation- ally such as e.g. Gaussian elimination. A strategy for addressing vector- ization along the same lines is also given.FCT, Mondrian Project funded by contract PTDC/EIA-CCO/108302/2008
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