Random Integer Lattice Generation via the Hermite Normal Form

Lattices used in cryptography are integer lattices. Defining and generating a “random integer lattice” are interesting topics. A generation algorithm for a random integer lattice can be used to serve as a random input of all the lattice algorithms. In this paper, we recall the definition of the random integer lattice given by G. Hu et al. and present an improved generation algorithm for it via the Hermite normal form. It can be proven that with probability ≥0.99, this algorithm outputs an n-dim random integer lattice within O(n2) operations.

Improved approximate rips filtrations with shifted integer lattices and cubical complexes

Rips complexes are important structures for analyzing topological features of metric spaces. Unfortunately, generating these complexes is expensive because of a combinatorial explosion in the complex size. For n points in $$\mathbb {R}^d$$ R d , we present a scheme to construct a 2-approximation of the filtration of the Rips complex in the $$L_\infty $$ L ∞ -norm, which extends to a $$2d^{0.25}$$ 2 d 0.25 -approximation in the Euclidean case. The k-skeleton of the resulting approximation has a total size of $$n2^{O(d\log k +d)}$$ n 2 O ( d log k + d ) . The scheme is based on the integer lattice and simplicial complexes based on the barycentric subdivision of the d-cube. We extend our result to use cubical complexes in place of simplicial complexes by introducing cubical maps between complexes. We get the same approximation guarantee as the simplicial case, while reducing the total size of the approximation to only $$n2^{O(d)}$$ n 2 O ( d ) (cubical) cells. There are two novel techniques that we use in this paper. The first is the use of acyclic carriers for proving our approximation result. In our application, these are maps which relate the Rips complex and the approximation in a relatively simple manner and greatly reduce the complexity of showing the approximation guarantee. The second technique is what we refer to as scale balancing, which is a simple trick to improve the approximation ratio under certain conditions.

Hankel determinants of a Sturmian sequence

<abstract><p>Let $ \tau $ be the substitution $ 1\to 101 $ and $ 0\to 1 $ on the alphabet $ \{0, 1\} $. The fixed point of $ \tau $ obtained starting from 1, denoted by $ {\bf{s}} $, is a Sturmian sequence. We first give a characterization of $ {\bf{s}} $ using $ f $-representation. Then we show that the distribution of zeros in the determinants induces a partition of integer lattices in the first quadrant. Combining those properties, we give the explicit values of the Hankel determinants $ H_{m, n} $ of $ {\bf{s}} $ for all $ m\ge 0 $ and $ n\ge 1 $.</p></abstract>

BEAM or MFA I inspired Nv Neurons using SPI and a MCU for line and line based polygon detection.

A companion publication to an opamp based Nv neuron architecture paper, this publication explores the use of inexpensive mouse optical sensors for shape recognition as polygons from line recognition networks, in sensor and two motor fusion in a TOMU/WOMU circuit using the SPI bus and a master -slave architecture. Lie Computability, is defined on discrete Tensor architectures, similar to computation on fields, in future work, field computing is proven to have the same complexity as integer lattices, though Lie Lattices embeddings in integer and complex lattices, proving MFA I and II architectures are equivalent in complexity, in both analog and digital worlds. Keywords: Tensor Flow, Tensor Architectures, Unsupervised Learning, Emergent A.I , procedural A.I, MFA I and II architectures, neuromodulation, SoC , TOMU/WOMU, SPI bus. What: We consider inexpensive 18 by 18 matrix 64 gray levels SPI interface, based photodetector components of optical mice. In this paper we consider the use of the SPI interface for the use of a master slave system of interface of an MCU to the optic processor for creating of BEAM circuitry using inexpensive MCU circuitry, such as the TOMU/WOMU. How: MFA I and MFA II architectures are fulfilled in both digital and analog circuitry, with a network architecture defined by a tensor notation, as described in a companion paper. Why: A digital fulfilment of a tensor architecture is defined and compared to Lego Mindstorm based deep learning and procedural algorithms for semantic segmentation and classification algorithms.

BEAM or MFA I inspired Nv Neurons using SPI and a MCU for line and line based polygon detection.

A companion publication to an opamp based Nv neuron architecture paper, this publication explores theuse of inexpensive mouse optical sensors for shape recognition as polygons from line recognitionnetworks, in sensor and two motor fusion in a TOMU/WOMU circuit using the SPI bus and a master-slave architecture. Lie Computability, is defined on discrete Tensor architectures, similar to computationon fields, in future work, field computing is proven to have the same complexity as integer lattices,though Lie Lattices embeddings in integer and complex lattices, proving MFA I and II architectures areequivalent in complexity, in both analog and digital worlds.Keywords: Tensor Flow, Tensor Architectures, Unsupervised Learning, Emergent A.I , procedural A.I,MFA I and II architectures, neuromodulation, MCU, SoC , TOMU/WOMU, SPI bus.What:We consider inexpensive 18 by 18 matrix 64 gray levels SPI interface, based photodetector componentsof optical mice. In this paper we consider the use of the SPI interface for the use of a master slave systemof interface of an MCU to the optic processor for creating of BEAM circuitry using inexpensive MCUcircuitry, such as the TOMU/WOMU.How:MFA I and MFA II architectures are fulfilled in both digital and analog circuitry, with a networkarchitecture defined by a tensor notation, as described in a companion paper.Why:A digital fulfilment of a tensor architecture is defined and compared to Lego Mindstorm based deeplearning and procedural algorithms for semantic segmentation and classification algorithms.