Ternary Representation Research Articles

Multidimensional Spectral Phasors of LAURDAN's Excitation-Emission Matrices: The Ultimate Sensor for Lipid Phases?

The impact of lipid diversity on the lateral organization of biological membranes remains a topic of debate. While the existence of domains in lamellar membranes is well-established, the nonlamellar phases occurring in biological systems are less explored due to technical constraints. Here, we present the measurement of the excitation-emission matrices (EEM) of LAURDAN in several lipid structures. LAURDAN is a fluorescence probe widely used for characterizing lipid assemblies. The EEMs were analyzed by multidimensional spectral phasors (MdSP), an approach that seizes information from both the excitation and emission spectra. We developed a computer algorithm to construct EEM data based on a model for LAURDAN's photophysics. The MdSP calculated from the simulated EEMs reveals that all feasible possibilities lie inside a universal triangle in the phasor's plot. We use this triangle to propose a ternary representation for the phasors, allowing a better assessment of LAURDAN's surroundings in terms of hydration, water mobility, and local electronic environment. Building upon this foundation, we constructed a theoretical "phase map" that can assess both lamellar and nonlamellar membranes. We thoroughly validated this theory using well-known lipid mixtures under different phase-state conditions and enzymatically generated systems. Our results confirm that the use of MdSP is a powerful tool for obtaining quantitative information on both lamellar and nonlamellar structures. This study not only advances our understanding of the impact of lipid diversity on membrane organization but also provides a robust and general framework for the assessment of fluorescence properties that can be further extended to other probes.

Lossless text compression by means of binary-coded ternary number representation

A lossless data compression code based on a binary-coded ternary number representation is investigated. The code is complete, universal, synchronizable, and supports a direct search in a compressed file. A simple monotonous encoding and very fast decoding algorithms are constructed owing to code properties. Experiments show that in natural language text compression, the new code outperforms byte-aligned codes SCDC and RPBC either in compression ratio or in decoding speed.

An Event-Based Neural Compressive Telemetry With >11× Loss-Less Data Reduction for High-Bandwidth Intracortical Brain Computer Interfaces.

Intracortical brain-computer interfaces offer superior spatial and temporal resolutions, but face challenges as the increasing number of recording channels introduces high amounts of data to be transferred. This requires power-hungry data serialization and telemetry, leading to potential tissue damage risks. To address this challenge, this paper introduces an event-based neural compressive telemetry (NCT) consisting of 8 channel-rotating Δ-ADCs, an event-driven serializer supporting a proposed ternary address event representation protocol, and an event-based LVDS driver. Leveraging a high sparsity of extracellular spikes and high spatial correlation of the high-density recordings, the proposed NCT achieves a compression ratio of >11.4×, while consumes only 1 µW per channel, which is 127× more efficient than state of the art. The NCT well preserves the spike waveform fidelity, and has a low normalized RMS error <23% even with a spike amplitude down to only 31 µV.

Efficient realization of quantum balanced ternary reversible multiplier building blocks: A great step towards sustainable computing

Nowadays, quantum computing plays a significant role in reducing the execution time in complicated computations. There are different reversible algorithms for quantum computation and many quantum circuits theoretically designed until now. These designs are only built from reversible gates because reversibility is a necessary logic in quantum circuits. Additionally, ternary representation can allow designing a circuit with fewer inputs and outputs, leading to a considerable decline in size or number of qutrits. Although other papers in quantum ternary reversible logic are in unbalanced representation, this paper, for the first time, considers balanced ternary advantages to achieve a more efficient design for quantum multipliers as the main component in arithmetic blocks. Six criteria are introduced to show how balanced representation increases the overall performance. In this work, two general balanced ternary adder blocks (full-adder and half-adder) and two balanced ternary partial product generators (BTPPGs) are proposed, and also a 2 × 2 and a 3 × 3 balanced ternary multipliers built from these components. The quantum costs of proposed balanced ternary multipliers enhanced 17% for a 2 × 2 multiplier and 34% for a 3 × 3 multiplier.

Mathematical self-determination theory I: Real representation

In two parts, MSDT1 this paper and MSDT2 the follow-up paper, we treat the topic of mathematical self-determination theory. MSDT1 considers the real representation, MSDT2 the affine space representation. The aim of the two papers is to lay the mathematical foundations of self-determination motivation theory. Self-determination theory was proposed by Deci and Ryan, which is a popular theory of motivation. The fundamental concepts are extrinsic and intrinsic motivation, amotivation, their type of regulation, locus of causality, and especially, self-determination. First, we give a geometric description of its concepts for the regulated case (no amotivation), as the unit 1-simplex. Thereby, we derive a symmetric definition of self-determination. Second, we extend the geometric description to the regulated and unregulated case, based on a more general ternary model, in internal motivation, external motivation, and amotivation. We define gradations of amotivation (and motivation), as 1-simplexes parallel to the unit 1-simplex. The ternary representation implies the types of strong, weak, and general self-determination, as partial orders on the motivation space. Third, we study the order, lattice, and algebraic properties of self-determination. In a version of polar coordinates, strong self-determination turns out to be a complete lattice on angular line segments, weak self-determination is a complete lattice on radial line segments, and general self-determination entails a complete lattice on the entire motivation space. In addition, the modified polar coordinates are employed to obtain necessary and sufficient conditions for strong, weak, and general self-determination. We propose measures for the strength of an ordinal dependency in self-determination, which are partial metrics on the motivation space.

A Digital Processing in Memory Architecture Using TCAM for Rapid Learning and Inference Based on a Spike Location Dependent Plasticity

In this paper, we present a digital processing in memory (DPIM) configured as a stride edge-detection search frequency neural network (SE-SFNN) which is trained through spike location dependent plasticity (SLDP), a learning mechanism reminiscent of spike timing dependent plasticity (STDP). This mechanism allows for rapid online learning as well as a simple memory-based implementation. In particular, we employ a ternary data scheme to take advantage of ternary content addressable memory (TCAM). The scheme utilizes a ternary representation of the image pixels, and the TCAMs are used in a two-layer format to significantly reduce the computation time. The first layer applies several filtering kernels, followed by the second layer, which reorders the pattern dictionaries of TCAMs to place the most frequent patterns at the top of each supervised TCAM dictionary. Numerous TCAM blocks in both layers operate in a massively parallel fashion using digital ternary values. There are no complicated multiply operations performed, and learning is performed in a feedforward scheme. This allows rapid and robust learning as a trade-off with the parallel memory block size. Furthermore, we propose a method to reduce the TCAM memory size using a two-tiered minor to major promotion (M2MP) of frequently occurring patterns. This reduction scheme is performed concurrently during the learning operation without incurring a preconditioning overhead. We show that with minimal circuit overhead, the required memory size is reduced by 84.4%, and the total clock cycles required for learning also decrease by 97.31 % while the accuracy decreases only by 1.12%. We classified images with 94.58% accuracy on the MNIST dataset. Using a 100 MHz clock, our simulation results show that the MNIST training takes about 6.3 ms dissipating less than 4 mW of average power. In terms of inference speed, the trained hardware is capable of processing 5,882,352 images per second.

Fundamentals of metric theory of real numbers in their $\overline{Q_3}$-representation

In the paper we were studied encoding of fractional part of a real number with an infinite alphabet (set of digits) coinciding with the set of non-negative integers. The geometry of this encoding is generated by $Q_3$-representation of real numbers, which is a generalization of the classical ternary representation. The new representation has infinite alphabet, zero surfeit and can be efficiently used for specifying mathematical objects with fractal properties.&#x0D; We have been studied the functions that store the "tails" of $\overline{Q_3}$-representation of numbers and the set of such functions,some metric problems and some problems of probability theory are connected with $\overline{Q_3}$-representation.

A key elements influence discovery scheme based on ternary association graph and representation learning

Key elements refer to the elements that play a crucial role in disseminating information in social networks. The influence discovery of key elements can guide a series of works, such as public opinion control, user recommendation, and marketing promotion. Recently, there have been many studies on the influence of elements, but at present, many methods focus on either the influence discovery of key elements of different types or the dynamic influence discovery of a certain type of element alone, and rarely consider the combination of the two. Therefore, this study proposes a key element discovery algorithm based on a ternary association graph and representation learning, which can detect the influence of paths, users, and user groups. Additionally, the changes of different types of key elements can be analyzed according to the influence of elements in each stage of topic communication.

Hybrid Encoding Scheme for AMBTC Compressed Images Using Ternary Representation Technique

Absolute moment block truncated coding (AMBTC) is a lossy image compression technique aiming at low computational cost, and has been widely studied. Previous studies have investigated the performance improvement of AMBTC; however, they often over describe the details of image blocks during encoding, causing an increase in bitrate. In this paper, we propose an efficient method to improve the compression performance by classifying image blocks into flat, smooth, and complex blocks according to their complexity. Flat blocks are encoded by their block means, while smooth blocks are encoded by a pair of adjusted quantized values and an index pointing to one of the k representative bitmaps. Complex blocks are encoded by three quantized values and a ternary map obtained by a clustering algorithm. Ternary indicators are used to specify the encoding cases. In our method, the details of most blocks can be retained without significantly increasing the bitrate. Experimental results show that, compared with prior works, the proposed method achieves higher image quality at a better compression ratio for all of the test images.

Living Donor-Recipient Pair Matching for Liver Transplant via Ternary Tree Representation With Cascade Incremental Learning.

Visual understanding of liver vessels anatomy between the living donor-recipient (LDR) pair can assist surgeons to optimize transplant planning by avoiding non-targeted arteries which can cause severe complications. We propose to visually analyze the anatomical variants of the liver vessels anatomy to maximize similarity for finding a suitable Living Donor-Recipient (LDR) pair. Liver vessels are segmented from computed tomography angiography (CTA) volumes by employing a cascade incremental learning (CIL) model. Our CIL architecture is able to find optimal solutions, which we use to update the model with liver vessel CTA images. A novel ternary tree based algorithm is proposed to map all the possible liver vessel variants into their respective tree topologies. The tree topologies of the recipient's and donor's liver vessels are then used for an appropriate matching. The proposed algorithm utilizes a set of defined vessel tree variants which are updated to maintain the maximum matching options by leveraging the accurate segmentation results of the vessels derived from the incremental learning ability of the CIL. We introduce a novel concept of in-order digital string based comparison to match the geometry of two anatomically varied trees. Experiments through visual illustrations and quantitative analysis demonstrated the effectiveness of our approach compared to state-of-the-art.

CNFET based design of unbalanced ternary circuits using efficient shifting literals

Multi-valued logic design provides the benefit of increased integration by reducing the interconnections which is the main source of power dissipation in the VLSI chip. As compared to conventional binary logic representation, ternary logic representation helps to represent more amount of information over the same number of digits. Hence this work presents the design of ternary adder, subtractor, multiplier and parallel adder/subtractor with enhanced performance in carbon nanotube field effect transistor (CNFET) technology. The addition and subtraction operations are set into a single unit, exploiting the advantage of ternary logic to increase the processing capability of the module. The presence of symmetricity among single shift and dual shift operators while realizing addition/subtraction operation further favors the combing of adder/subtractor blocks. Then 4-stage parallel adder/subtractor is designed. Further single-trit and multi-trit multiplier circuits are also implemented by employing the proposed adder and multiplier units. All the proposed arithmetic designs are derived from the proposed ternary multiplexer and shifting operator circuits. To test the efficiency of proposed circuits, the simulations are conducted on 32 nm CNFET technology using Synopsis HSPICE simulator. In the proposed methodology reduction in power consumption of 38% for combined adder/subtraction block and 40% for parallel adder/subtractor unit is obtained. Furthermore, for the multi trit adder/subtractor and multiplier designs, the maximum power delay product (PDP) improvement up to 66% is achieved.

Harmonious Coexistence of Structured Weight Pruning and Ternarization for Deep Neural Networks

Deep convolutional neural network (DNN) has demonstrated phenomenal success and been widely used in many computer vision tasks. However, its enormous model size and high computing complexity prohibits its wide deployment into resource limited embedded system, such as FPGA and mGPU. As the two most widely adopted model compression techniques, weight pruning and quantization compress DNN model through introducing weight sparsity (i.e., forcing partial weights as zeros) and quantizing weights into limited bit-width values, respectively. Although there are works attempting to combine the weight pruning and quantization, we still observe disharmony between weight pruning and quantization, especially when more aggressive compression schemes (e.g., Structured pruning and low bit-width quantization) are used. In this work, taking FPGA as the test computing platform and Processing Elements (PE) as the basic parallel computing unit, we first propose a PE-wise structured pruning scheme, which introduces weight sparsification with considering of the architecture of PE. In addition, we integrate it with an optimized weight ternarization approach which quantizes weights into ternary values ({-1,0,+1}), thus converting the dominant convolution operations in DNN from multiplication-and-accumulation (MAC) to addition-only, as well as compressing the original model (from 32-bit floating point to 2-bit ternary representation) by at least 16 times. Then, we investigate and solve the coexistence issue between PE-wise Structured pruning and ternarization, through proposing a Weight Penalty Clipping (WPC) technique with self-adapting threshold. Our experiment shows that the fusion of our proposed techniques can achieve the best state-of-the-art ∼21× PE-wise structured compression rate with merely 1.74%/0.94% (top-1/top-5) accuracy degradation of ResNet-18 on ImageNet dataset.

VLSI Implementation of Digital Filter using Novel RTSD Adder and Booth Multiplier

Finite Impulse Response (FIR) filters are most important element in signal processing and communication. Area and speed optimization are the essential necessities of FIR filter design. This work looks at the design of Finite Impulse Response (FIR) filters from an arithmetic perspective. Since the fundamental arithmetic operations in the convolution equations are addition and multiplication, they are the objectives of the design analysis. For multiplication, Booth encoding is utilized in order to lessen the quantity of partial products. Consequently, considering carry-propagation free addition strategies should improve the addition operation of the filter. The redundant ternary signed-digit (RTSD) number framework is utilized to speedup addition in the filter. The redundant ternary representation utilizes more bits than required to denote the single binary digit because of which most numbers have several representations. This special behavior of RTSD allows the addition along with the absence of typical carry propagation. Xilinx ISE design suite 14.5 is used for the design and validation of proposed method. From the implementation result, the proposed design of FIR filter is compared with other conventional techniques to show the better performance by means of power, area and delay.

Fast and energy-efficient FPGA realization of RNS reverse converter for the ternary 3-moduli set {3n–2, 3n–1, 3n}

Ternary number representation has been known as one of the efficient methods for the implementation of digital systems. Employing residue number system (RNS) in ternary digital signal processing (DSP) applications could offer the advantages of ternary number representation along with further privileges such as improving parallelism and fault tolerance. Efficacious design of reverse converter as a crucial block of residue number system has always been of great interest to researchers. In this paper, an efficient reverse converter for ternary RNS with moduli set {3n–2, 3n–1, 3n}, based on the mixed-radix conversion (MRC) algorithm is presented. The proposed algorithm reduces the number of adders with large bit-width and the dependencies between consecutively process blocks. The employed technique in the proposed algorithm and the reverse converter architecture improves the delay and area on the FPGA Virtex Ultrascale + family platform in 14 nm/16 nm FinFET technology as compared to the related works.

Ternary representation of N (N = 1 or 2)-input and 1-output algorithmic self-assembly demonstrated by DNA

Deoxyribonucleic acid (DNA) is effective for molecular computation because of its high energy efficiency, high information density, and parallel-computing capability. Although logic implementation using DNA molecules is well established in binary systems (base value of 2) via decoration of hairpin structures on DNA duplexes, systems with base values of >2 (e.g. 3, corresponding to a ternary system) are rarely discussed owing to the complexity of the design and the experimental difficulties with DNA. In this study, DNA rule tiles that participate to form algorithmic DNA crystals exhibiting the ternary representation of an N (N = 1 or 2)-input and 1-output algorithmic assembly are conceived. The number of possible algorithmic patterns is in the ternary N-input and 1-output logic gate. Thus, the number of possible rules is 27 (=33) for a 1-input and 1-output algorithmic logic gate and 19 638 (=39) for a 2-input and 1-output algorithmic logic gate. Ternary bit information (i.e. 0-, 1-, and 2-bit) is encoded on rule tiles without hairpins and with short and long hairpins. We construct converged, line-like, alternating, and commutative patterns by implementing specific rules (TR00, TR05, TR07, and TR15, respectively) for the 1-input and 1-output gate and an ascending line-like pattern (with the rule of TR3785) for the 2-input and 1-output gate. Specific patterns generated on ternary-representing rule-embedded algorithmic DNA crystals are visualized via atomic force microscopy, and the errors during the growth of the crystals are analyzed (average error rates obtained for all experimental data are <4%). Our method can easily be extended to a system having base values of >3.

Toward efficient implementation of basic balanced ternary arithmetic operations in CNFET technology

The balanced ternary representation allows for significant performance improvement for ternary systems. Immaturity of the semiconductor industry in providing an applicable device for ternary infrastructures during the past decades is an enormous obstacle to actualize the ternary circuit realization. Among the promising devices for this purpose, carbon nanotube field effect transistor (CNFET) is qualified with outstanding characteristics which make it suitable for implementing the ternary logic functions. In this paper, for the first time, we present and evaluate the basic ternary arithmetic circuits in balanced representation based on the CNFET technology. Experimental evaluation of reviewed designs in both balanced and unbalanced representation indicates the superiority of the balanced representation in the design of ternary arithmetic blocks.

Ternary Functions Design Using Memristive Threshold Logic

Memristive threshold logic (MTL) concept is emerged in many circuits to enable high-performance systems in terms of power, energy, area, and delay. This paper proposes a systematic method for building two-bit ternary number functions based on the MTL concept. The proposed method is applied to build the basic ternary arithmetic operations. The implementation of two-bit adder and multiplier is presented in the unbalanced ternary number representation. The proposed designs are verified by using VTEAM memristor and Stanford CNTFET transistor models. Finally, a comparison between the proposed circuits and related work presented in this paper is discussed. It shows that the area in case of the ternary adder is reduced by 30% and 76% and in case of the ternary multiplier by considering that memristors can be stacked above the transistors. In addition, this reduction in the number of transistors reduces the circuit static power and hence improving the overall ternary circuits performance.

ZnTe semiconductor nanoparticles: A chemical approach of the mechanochemical synthesis

To have a better understanding on how ZnTe semiconductor nanoparticles were obtained by high-energy milling under composition, pressure and temperature vial conditions reached during milling, a molar Zn–Te–O ternary representation was disclosed. Through this approach, all the bulk and surface chemical transitions were revealed as: i) stable, ii) stoichiometric and iii) metastable or non-stoichiometric reaction products in the Zn–Te–O molar system. Surface studies (via X-ray photoelectron spectroscopy and high-resolution transmission electron microscopy) and bulk analyses (X-ray diffraction and secondary ion mass spectrometry) were carried out to find out the mechanism of formation of the ZnTe semiconductor nanoparticles. The results showed that the as-milled ZnTe nanoparticles were polycrystalline – with a slight presence of amorphous phases – and chemically homogeneous (according to elemental chemical mapping of Zn K-line and Te L-line). Thus, the way in which the oxygen chemical potential was decreased from Zn2Te3O8 to ZnTe nanoparticles was proposed. Consequently, predicted transitions in the ternary molar representation were consistent with experimental results. A blue shift of the ZnTe nanoparticles (at 10 h milling time) was observed by optical measurements. It implied an enhancement in its direct band gap, attributed to the presence of nanoparticles and coarse particles with both, embedded quantum dots and nanoparticles.

Wythoff partizan subtraction

We introduce a class of normal-play partizan games, called Complementary Subtraction. These games are instances of Partizan Subtraction where we take any set A of positive integers to be Left’s subtraction set and let its complement be Right’s subtraction set. In wythoff partizan subtraction we take the set A and its complement B from wythoff nim, as the two subtraction sets. As a function of the heap size, the maximum size of the canonical forms grows quickly. However, the value of the heap is either a number or, in reduced canonical form, a switch. We find the switches by using properties of the Fibonacci word and standard Fibonacci representations of integers. Moreover, these switches are invariant under shifts by certain Fibonacci numbers. The values that are numbers, however, are distinct, and we can find their binary representation in polynomial time using a representation of integers as sums of Fibonacci numbers, known as the ternary (or “the even”) Fibonacci representation.

Formulation and characterization of blended alkali-activated materials based on flash-calcined metakaolin, fly ash and GGBS

Studies performed on alkali-activated materials (AAMs) in recent years reveal that they could be an alternative to ordinary Portland cement (OPC). The main precursors used are flash-calcined metakaolin (MK), fly ash (FA), and ground granulated blast furnace slag (GGBS). Usually, only one precursor is used to produce an AAM. The aims of this study were to measure and compare the performances of pure (one precursor) and blended AAMs (two precursors). Alkaline solution is added to activate the precursors and the influence of the activation rate was also investigated by changing the activator amount (15% and 25% of alkaline solution dry extract). Twenty-six different mixes were studied and are presented in a ternary representation. The mixtures were characterized in the fresh state (initial setting time) and the main properties useful for building applications (shrinkage, mass loss, and compressive strength) were recorded and analysed. In contrast to the situation in OPC systems, shrinkage and mass loss in AAMs were not directly correlated. The high shrinkage values observed for pure MK AAMs and pure GGBS AAMs were significantly reduced when they were associated in appropriate proportions. The association of MK and GGBS also proved very interesting regarding early age compressive strength.