On Classification of k-Dimension Paths in n-Cube

The shortest k-dimension paths (k-paths) between vertices of n-cube are considered on the basis a bijective mapping of k-faces into words over a finite alphabet. The presentation of such paths is proposed as (n - k + 1)×n matrix of characters from the same alphabet. A classification of the paths is founded on numerical invariant as special partition. The partition consists of n parts, which correspond to columns of the matrix.

Research on enumeration problem of pattern-avoiding permutations

The problem of relevant enumeration with pattern-avoiding permutations is a significant topic in enumerative combinatorics and has wide applications in physics,chemistry,and computer science.This paper summarizes the relevant conclusions of the enumeration of pattern-avoiding permutations on the nelement symmetric group Sn,alternating permutations,Dumont permutations,Ballot permutations,and inversion sequences.It also introduces relevant research results on avoiding vincular patterns and barred patterns in S_(n).

Practical construction of globally injective parameterizations with positional constraints

We propose a novel method to compute globally injective parameterizations with arbitrary positional constraints on disk topology meshes.Central to this method is the use of a scaffold mesh that reduces the globally injective constraint to a locally flipfree condition.Hence,given an initial parameterized mesh containing flipped triangles and satisfying the positional constraints,we only need to remove the flips of a overall mesh consisting of the parameterized mesh and the scaffold mesh while always meeting positional constraints.To successfully apply this idea,we develop two key techniques.Firstly,an initialization method is used to generate a valid scaffold mesh and mitigate difficulties in eliminating flips.Secondly,edgebased remeshing is used to optimize the regularity of the scaffold mesh containing flips,thereby improving practical robustness.Compared to state-of-the-art methods,our method is much more robust.We demonstrate the capability and feasibility of our method on a large number of complex meshes.

A GNERAL BIJECTIVE ALGORITHM FOR INCREASING TREES

The main concern of this paper is with a bijective approach to various classes ofincreasing trees. We discover an increasing tree counterpart of the decomposition algorithmfor Schroder trees. This bijection has probably reached its full generality for decomposingincreasing trees. As a special case of our algorithm, we provide a solution to a problemconcerning the enumeration of plane trees by the net number of inversions.

Auslander’s defect formula and a commutative triangle in an exact category

We prove Auslander’s defect formula in an exact category,and obtain a commutative triangle involving the Auslander bijections and the generalized Auslander Reiten duality.

Indexing structures for the PLS blockchain

This paper studies known indexing structures from a new point of view:minimisation of data exchange between an IoT device acting as a blockchain client and the blockchain server running a protocol suite that includes two Guy Fawkes protocols,PLS and SLVP.The PLS blockchain is not a cryptocurrency instrument;it is an immutable ledger offering guaranteed non-repudiation to low-power clients without use of public key crypto.The novelty of the situation is in the fact that every PLS client has to obtain a proof of absence in all blocks of the chain to which its counterparty does not contribute,and we show that it is possible without traversing the block's Merkle tree.We obtain weight statistics of a leaf path on a sparse Merkle tree theoretically,as our ground case.Using the theory we quantify the communication cost of a client interacting with the blockchain.We show that large savings can be achieved by providing a bitmap index of the tree compressed using Tunstall's method.We further show that even in the case of correlated access,as in two IoT devices posting messages for each other in consecutive blocks,it is possible to prevent compression degradation by re-randomising the IDs using a pseudorandom bijective function.We propose a low-cost function of this kind and evaluate its quality by simulation,using the avalanche criterion.

Indexing structures for the PLS blockchain

This paper studies known indexing structures from a new point of view:minimisation of data exchange between an IoT device acting as a blockchain client and the blockchain server running a protocol suite that includes two Guy Fawkes protocols,PLS and SLVP.The PLS blockchain is not a cryptocurrency instrument;it is an immutable ledger offering guaranteed non-repudiation to low-power clients without use of public key crypto.The novelty of the situation is in the fact that every PLS client has to obtain a proof of absence in all blocks of the chain to which its counterparty does not contribute,and we show that it is possible without traversing the block’s Merkle tree.We obtain weight statistics of a leaf path on a sparse Merkle tree theoretically,as our ground case.Using the theory we quantify the communication cost of a client interacting with the blockchain.We show that large savings can be achieved by providing a bitmap index of the tree compressed using Tunstall’s method.We further show that even in the case of correlated access,as in two IoT devices posting messages for each other in consecutive blocks,it is possible to prevent compression degradation by re-randomising the IDs using a pseudorandom bijective function.We propose a low-cost function of this kind and evaluate its quality by simulation,using the avalanche criterion.

G-stable support τ-tilting modules

Motivated by T-tilting theory developed by T. Adachi, O. Iyama, I. Reiten, for a finite-dimensional algebra A with action by a finite group G, we introduce the notion of G-stable support τ-tilting modules. Then we establish bijections among G-stable support τ-tilting modules over ∧, G-stable two-term silting complexes in the homotopy category of bounded complexes of finitely generated projective ∧-modules, and G-stable functorially finite torsion classes in the category of finitely generated left ∧-modules. In the case when ∧ is the endomorphism of a G-stable cluster-tilting object T over a Horn-finite 2-Calabi- Yau triangulated category L with a G-action, these are also in bijection with G-stable cluster-tilting objects in L. Moreover, we investigate the relationship between stable support τ-tilitng modules over ∧ and the skew group algebra ∧G.

An Operator on Ascent Sequences

We solve two problems about ascent sequences： how to get the ascent sequence of the reflection of A with respect to its antidiagonal for a matrix A ∈ Intn and its ascent sequences, and how to determine the ascent sequence of A＋B for kxk matrices A ∈ Int, and B ∈ Intm. We give the other definition of ascent se- quence and get M-sequence. For the first question, we define M-sequence of A and rewrite the ascent sequences as another form We build the bijection between M-sequences and ascent sequences and prove that our bijection is well-defined. For the second ques- tion, we define an operation on M-sequences. On the basis of the operation and the bijections, we get the ascent sequences of the sum of two matrices.

Parallel processing via carbon field emission-based controlled switching of regular bijective nano systolic networks,part 1:basics

Purpose-The purpose of this paper is to introduce new implementations for parallel processing applications using bijective systolic networks and the corresponding carbon-based field emission controlled switching.The developed implementations are performed in the reversible domain to perform the required bijective parallel computing,where the implementations for parallel computations that utilize the presented field-emission controlled switching and their corresponding m-ary(many-valued)extensions for the use in nano systolic networks are introduced.The first part of the paper presents important fundamentals with regards to systolic computing and carbon-based field emission that will be utilized in the implementations within the second part of the paper.Design/methodology/approach-The introduced systolic systems utilize recent findings in field emission and nano applications to implement the functionality of the basic bijective systolic network.This includes many-valued systolic computing via field emission techniques using carbon-based nanotubes and nanotips.The realization of bijective logic circuits in current and emerging technologies can be very important for various reasons.The reduction of power consumption is a major requirement for the circuit design in future technologies,and thus,the new nano systolic circuits can play an important role in the design of circuits that consume minimal power for future applications such as in low-power signal processing.In addition,the implemented bijective systems can be utilized to implement massive parallel processing and thus obtaining very high processing performance,where the implementation will also utilize the significant size reduction within the nano domain.The extensions of implementations to field emission-based many-valued systolic networks using the introduced bijective nano systolic architectures are also presented.Findings-Novel bijective systolic architectures using nano-based field emission implementations are introduced in this paper,and the implementation using the general scheme of many-valued computing is presented.The carbon-based field emission implementation of nano systolic networks is also introduced.This is accomplished using the introduced field emission carbon-based devices,where field emission from carbon nanotubes and nano-apex carbon fibers is utilized.The implementations of the many-valued bijective systolic networks utilizing the introduced nano-based architectures are also presented.Originality/value-The introduced bijective systolic implementations form new important directions in the systolic realizations using the newly emerging nano-based technologies.The 2-to-1 multiplexer is a basic building block in“switch logic,”where in switch logic,a logic circuit is realized as a combination of switches rather than a combination of logic gates as in the gate logic,which proves to be less costly in synthesizing multiplexer-based wide variety of modern circuits and systems since nano implementations exist in very compact space where carbon-based devices switch reliably using much less power than silicon-based devices.The introduced implementations for nano systolic computation are new and interesting for the design in future nanotechnologies that require optimal design specifications of minimum power consumption and minimum size layout such as in low-power control of autonomous robots and in the adiabatic low-power very-large-scale-integration circuit design for signal processing applications.

Parallel processing via carbon field emission-based controlled-switching of regular bijective nano systolic networks,Part II Architectural implementation

Purpose–The purpose of this paper is to introduce new implementations for parallel processing applications using bijective systolic networks and their corresponding carbon-based field emission controlled switching.The developed implementations are performed in the reversible domain to perform the required bijective parallel computing,where the implementations for parallel computations that utilize the presented field-emission controlled switching and their corresponding many-valued(m-ary)extensions for the use in nano systolic networks are introduced.The second part of the paper introduces the implementation of systolic computing using two-to-one controlled switching via carbon-based field emission that were presented in the first part of the paper,and the computational extension to the general case of many-valued(m-ary)systolic networks utilizing many-to-one carbon-based field emission is also introduced.Design/methodology/approach–The introduced systolic systems utilize recent findings in field emission and nano applications to implement the functionality of the basic bijective systolic network.This includes many-valued systolic computing via field-emission techniques using carbon-based nanotubes and nanotips.The realization of bijective logic circuits in current and emerging technologies can be very important for various reasons.The reduction of power consumption is a major requirement for the circuit design in future technologies,and thus,the new nano systolic circuits can play an important role in the design of circuits that consume minimal power for future applications such as in low-power signal processing.In addition,the implemented bijective systems can be utilized to implement massive parallel processing and thus obtaining very high processing performance,where the implementation will also utilize the significant size reduction within the nano domain.The extensions of implementations to field emission-based many-valued systolic networks using the introduced bijective nano systolic architectures are also presented.Findings–Novel bijective systolic architectures using nano-based field emission implementations are introduced in this paper,and the implementation using the general scheme of many-valued computing is presented.The carbon-based field emission implementation of nano systolic networks is also introduced.This is accomplished using the introduced field-emission carbon-based devices,where field emission from carbon nanotubes and nano-apex carbon fibersisutilized.The implementationsof the many-valued bijective systolic networks utilizing the introduced nano-based architectures are also presented.Practical implications–The introduced bijective systolic implementations form new important directions in the systolic realizations using the newly emerging nano-based technologies.The 2-to-1 multiplexer is a basic building block in“switch logic,”where in switch logic,a logic circuit is realized as a combination of switches rather than a combination of logic gates as in the gate logic,which proves to be less costly in synthesizing multiplexer-based wide variety of modern circuits and systems since nano implementations exist in very compact space where carbon-based devices switch reliably using much less power than silicon-based devices.The introduced implementations for nano systolic computation are new and interesting for the design in future nanotechnologies that require optimal design specifications of minimum power consumption and minimum size layout such as in low-power control of autonomous robots and in the adiabatic low-power VLSI circuit design for signal processing applications.Originality/value–The introduced bijective systolic implementations form new important directions in the systolic realizations utilizing the newly emerging nanotechnologies.The introduced implementations for nano systolic computation are new and interesting for the design in future nanotechnologies that require optimal design specifications of high performance,minimum power and minimum size.

Locally Inverse Semigroups with Inverse Transversals

Let S be a locally inverse semigroup with an inverse transversal S°.In this paper,we construct an amenable partial order on S by an R-cone.Conversely,every amenable partial order on S can be constructed in this way.We give some properties of a locally inverse semigroup with a Clifford transversal.In particular,if S is a locally inverse semigroup with a Clifford transversal,then there is an order-preserving bijection from the set of all amenable partial orders on S to the set of all R-cones of S.