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This paper presents the design, implementation, and validation of a novel robust control strategy based on variables transformation, i.e., transforming variables to obtain the triangular flat canonical normal form (TCNF) for nonlinear dynamic systems satisfying the flatness geometric property. A differential flatness, integrated with a geometrical transformation-based method, was developed to control an exoskeleton robot named Smart Robotic Exoskeleton (SREx). The SREx was designed to provide upper limb rehabilitation therapy to the stroke survivors. The proposed flatness control technique ensures continuous enhanced trajectory tracking and the system's high robustness during following a trajectory representing rehabilitation exercises. Furthermore, utilizing the Lyapunov theory, all signals were shown to be bounded in the closed-loop form. Experiments and comparative studies were carried out to validate the effectiveness of the proposed control scheme. Controlled experiments were further conducted using healthy subjects, maneuvering the SERx to provide rehabilitation exercise and to validate the control scheme in real-time.

Schur's inequality for the sum of products of the differences of real numbers states that for x,y,z,t≥0, xt(x−y)(x−z)+yt(y−z)(y−x)+zt(z−x)(z−y)≥0. In this paper, we study a generalization of this i...

On-shell amplitude methods have proven to be extremely efficient for calculating anomalous dimensions. We further elaborate on these methods to show that, by the use of an angular momentum decomposition, the one-loop anomalous dimensions can be reduced to essentially a sum of products of partial waves. We apply this to the SM EFT, and show how certain classes of anomalous dimensions have their origin in the same partial-wave coefficients. We also use our result to obtain a generic formula for the one-loop anomalous dimensions of nonlinear sigma models at any order in the energy expansion, and apply our method to gravity, where it proves to be very advantageous even in the presence of IR divergencies., 22 pages, 1 figure; v2: signs corrected and conventions clarified, other minor changes

Reliability block diagram (RBD) is a well-known, high-level abstract modeling method for calculating systems reliability. Increasing redundancy is the most important way for increasing Fault-tolerance and reliability of dependable systems. K-out-of-N (KooN) is one of the known redundancy models. The redundancy causes repeated events and increases the complexity of the computing system’s reliability, and researchers use techniques like factorization to overcome it. Current methods lead to the cumbersome formula that needs a lot of simplification to change in the form of Sum of the Products (SoP) in terms of reliabilities of its constituting components. In This paper, a technique for extracting simple formula for calculating the KooN system’s reliability in SoP form using the Venn diagram is presented. Then, the shortcoming of using the Venn diagram that is masking some joints events in the case of a large number of independent components is explained. We proposed the replacement of Lattice instead of Venn diagrams to overcome this weakness. Then, the Lattice of reliabilities that is dual of power set Lattice of components is introduced. Using the basic properties of Lattice of reliabilities and their inclusion relationships, we propose an algorithm for driving a general formula of the KooN system’s reliability in SoP form. The proposed algorithm gives the SoP formula coefficients by computing elements of the main diagonal and elements below it in a squared matrix. The computational and space complexity of the proposed algorithm is $\theta (\left ({n-k }\right)^{2}/2)$ that $n$ is the number of different components and $k$ denotes the number of functioning components. A lemma and a theorem are defined and proved as a basis of the proposed general formula for computing coefficients of the SoP formula of the KooN system. Computational and space complexity of computing all of the coefficients of reliability formula of KooN system using this formula reduced to $\theta (n-k)$ . The proposed formula is simple and is in the form of SoP, and its computation is less error-prone.

Given a vectorial function ${F}:\mathbb {F}_{2}^{{n}} \mapsto \mathbb {F}_{2}^{{m}}$ , the indicator $1_{{\mathcal {G}}_{{F}}}$ of its graph ${\mathcal{ G}}_{{F}}=\{({x},{F}({x})); {x}\in \mathbb {F}_{2}^{{n}}\}$ allows to express the algebraic degree of $F$ in a simple way. Exploiting the formula, obtained in a previous article, for the graph indicator of a composite function ${G}\circ {F}$ , that involves only a sum of products of $1_{{\mathcal {G}}_{{F}}}$ and $1_{{\mathcal {G}}_{{G}}}$ , we deduce the exact expression as well as bounds on the algebraic degree of ${G}\circ {F}$ , whose efficiency comes from the fact that the algebraic degree of the product of two Boolean functions is bounded above by the sum of their algebraic degrees, while for a composition, it is bounded above by their product. One of these bounds, that depends on the algebraic degrees of $G$ and $1_{{\mathcal {G}}_{{F}}}$ , is tight, general, simple, and most often efficient (for the case where it is not efficient, we give an improved bound, that is a little more complex). As far as we know, it is the first efficient upper bound ever found, that works without any condition on the vectorial functions. It provides a new criterion for the choice of S-boxes in block ciphers. It implies as a corollary a known bound assuming the divisibility of the Walsh transform values by a power of 2. It gives a better view why this latter bound works. All the bounds generalize to more than two functions and this represents also an improvement over the state of the art. When $F$ is a permutation, our expression of the algebraic degree of ${G}\circ {F}$ simplifies into a formula involving the algebraic degrees of the products of a coordinate function of $G$ and coordinate functions of ${F}^{-1}$ . This implies another known bound showing that the algebraic degree of ${F}^{-1}$ has more impact on that of ${G}\circ {F}$ than that of $F$ itself. Our approach by graph indicators gives a more complete explanation to this interesting fact. Our results include all the known efficient bounds as particular cases, and clarify the reasons why they work. We also deduce the exact expression of the algebraic degree of the composition of any number of functions, leading to a bound that is much more efficient than what we obtain by applying the known bound several times. We also obtain two bounds on the algebraic degree of ${G} \circ {F}$ , where $F$ is a permutation, given the divisibility by powers of 2 of some Walsh transform values of component functions of $F$ and their sums with a coordinate function of $G$ . We compare all the bounds of this kind obtained so far and show how they are complementary, and we study the generalizations of all (known and new) bounds of this kind to the composition of more than two functions.

In this paper, a matrix approach to the direct computation method for solving Fredholm integro-differential equations (FIDEs) of the second kind with degenerate kernels is presented. Our approach consists of reducing the problem to a set of linear algebraic equations by approximating the kernel with a finite sum of products and determining the unknown constants by the matrix approach. The proposed method is simple, efficient and accurate; it approximates the solutions exactly with the closed form solutions. Some problems are considered using maple programme to illustrate the simplicity, efficiency and accuracy of the proposed method.

Let $$\mathcal {A}$$ and $$\mathcal {B}$$ be two factor von Neumann algebras. In this paper, we proved that a bijective mapping $$\varPhi :\mathcal {A}\rightarrow \mathcal {B}$$ satisfies $$\varPhi (a\circ b-ba^{*})=\varPhi (a)\circ \varPhi (b)-\varPhi (b)\varPhi (a)^{*}$$ (where $$\circ $$ is the special Jordan product on $$\mathcal {A}$$ and $$\mathcal {B},$$ respectively), for all elements $$a,b\in \mathcal {A}$$ , if and only if $$\varPhi $$ is a $$*$$ -ring isomorphism. In particular, if the von Neumann algebras $$\mathcal {A}$$ and $$\mathcal {B}$$ are type I factors, then $$\varPhi $$ is a unitary isomorphism or a conjugate unitary isomorphism.

In this paper, the regularities of convolution of sequences c of Fibonacci numbers {Fn} generated by metallic means and the sum of products of two statistically independent sequences {Fi} and Jn=j∙sin(0.5π(n-j)) are investigated. It is shown that the known closed forms of sums for convolution and product are similar. Attention to the study of the convolution of two sequences of discrete data is associated with the use of this method for statistical signal processing. This problem involves calculating finite sums as discrete analogs of definite integrals. Such a problem is considered solved if the formula for the sum is expressed in a closed form as a function of its members and their number.

International audience; This paper deals with the assembly line balancing issue. The considered objective is to minimize the weighted sum of products’ cycle times. The originality of this objective is that it is the generalization of the cycle time minimization used in single-model lines (SALBP) to the multi-model case (MALBP). An optimization algorithm made of a heuristic and a tabu-search method is presented and evaluated through an experimental study carried out on several and various randomly generated instances for both the single and multiproduct cases. The returned solutions are compared to optimal solutions given by a mathematical model from the literature and to a proposed lower bound inspired from the classical SALBP bound. The results show that the algorithm is highperforming as the average relative gap between them is quite low for both problems.

The Chow rank of a form is the length of its smallest decomposition into a sum of products of linear forms. For a generic form, this corresponds to finding the smallest secant variety of the Chow variety which fills the ambient space. We determine the Chow rank of generic cubics and quaternary forms by proving nondefectivity of all involved secant varieties. The main new ingredient in our proof is the generalization of a technique by [Brambilla and Ottaviani, On the Alexander--Hirschowitz theorem, J. Pure Appl. Algebra, 2008] that consists of employing Terracini's lemma and Newton's backward difference formula to compute the dimensions of secant varieties of arbitrary projective varieties. Via this inductive construction, the proof of nondefectivity ultimately reduces to proving a number of base cases. These are settled via a computer-assisted proof because of the large dimensions of the spaces involved. The largest base case required in our proof consisted of computing the dimension of a vector space constructed from the $400$th secant variety of a degree-$82$ Chow variety embedded in $\mathbb{P}^{98769}$., 21 pages, 1 figure

Cumulants are a notion that comes from the classical probability theory, they are an alternative to a notion of moments. We adapt the probabilistic concept of cumulants to the setup of a linear space equipped with two multiplication structures. We present an algebraic formula which involves those two multiplications as a sum of products of cumulants. In our approach, beside cumulants, we make use of standard combinatorial tools as forests and their colourings. We also show that the resulting statement can be understood as an analogue of Leonov--Shiraev's formula. This purely combinatorial presentation leads to some conclusions about structure constant of Jack characters., 26 pages

We study limitations of polynomials computed by depth-2 circuits built over read-once formulas (ROFs). In particular: • We prove a 2 Ω( n ) lower bound for the sum of ROFs computing the 2 n -variate polynomial in VP defined by Raz and Yehudayoff [21]. • We obtain a 2 Ω(√ n ) lower bound on the size of Σ Π [ n 1/15 ] arithmetic circuits built over restricted ROFs of unbounded depth computing the permanent of an n × n matrix (superscripts on gates denote bound on the fan-in). The restriction is that the number of variables with + gates as a parent in a proper sub formula of the ROF has to be bounded by √ n . This proves an exponential lower bound for a subclass of possibly non-multilinear formulas of unbounded depth computing the permanent polynomial. • We also show an exponential lower bound for the above model against a polynomial in VP. • Finally, we observe that the techniques developed yield an exponential lower bound on the size of ΣΠ [ N 1/30 ] arithmetic circuits built over syntactically multi-linear ΣΠΣ [ N 1/4 ] arithmetic circuits computing a product of variable disjoint linear forms on N variables, where the superscripts on gates denote bound on the fan-in. Our proof techniques are built on the measure developed by Kumar et al. [14] and are based on a non-trivial analysis of ROFs under random partitions. Further, our results exhibit strengths and provide more insight into the lower bound techniques introduced by Raz [19].

This paper deals with canonical normal form approach for trajectory tracking control of a two degree-of-freedom helicopter system. The main objective is to stabilize beam of the helicopter with respect to pitch and yaw angles. Sufficient geometrical conditions are given for a class of nonlinear dynamical systems to transform the studied nonlinear systems into a triangular normal form and determine the flat outputs. Using this form two controllers are designed to control the position of the yaw and the pitch angles of the Helicopter. Simulation results are given to illustrate the effectiveness of the proposed control scheme.

Poincaré's work on the reduction of abelian integrals contains implicitly an algorithm for the expression of a theta function as a sum of products of theta functions of fewer variables in the presence of reduction . The aim of this paper is to give explicit formulations and reasonably complete proofs of Poincaré's results.

Let \(\mathcal{A}\) and \(\mathcal{B}\) be two factor von Neumann algebras. In this paper, we proved that a bijective mapping \(\Phi :\mathcal{A}\rightarrow \mathcal{B}\) satisfies \(\Phi (a\circ b+ba^{*})=\Phi (a)\circ \Phi (b)+\Phi (b)\Phi (a)^{*}\) (where \(\circ \) is the special Jordan product on \(\mathcal{A}\) and \(\mathcal{B},\) respectively), for all elements \(a,b\in \mathcal{A}\), if and only if \(\Phi \) is a \(\ast \)-ring isomorphism. In particular, if the von Neumann algebras \(\mathcal{A}\) and \(\mathcal{B}\) are type I factors, then \(\Phi \) is a unitary isomorphism or a conjugate unitary isomorphism.

In the Monte Carlo many-body perturbation (MC-MP) method, the conventional correlation-correction formula, which is a long sum of products of low-dimensional integrals, is first recast into a short sum of high-dimensional integrals over electron-pair and imaginary-time coordinates. These high-dimensional integrals are then evaluated by the Monte Carlo method with random coordinates generated by the Metropolis-Hasting algorithm according to a suitable distribution. The latter algorithm, while advantageous in its ability to sample nearly any distribution, introduces autocorrelation in sampled coordinates, which, in turn, increases the statistical uncertainty of the integrals and thus the computational cost. It also involves wasteful rejected moves and an initial "burn-in" step as well as displays hysteresis. Here, an algorithm is proposed that directly produces a random sequence of electron-pair coordinates for the same distribution used in the MC-MP method, which is free from autocorrelation, rejected moves, a burn-in step, or hysteresis. This direct-sampling algorithm is shown to accelerate second- and third-order Monte Carlo many-body perturbation calculations by up to 222% and 38%, respectively.

The Logarithmic Number System (LNS) is useful in applications that tolerate approximate computation, such as classification using multi-layer neural networks that compute nonlinear functions of weighted sums of inputs from previous layers. Supervised learning has two phases: training (find appropriate weights for the desired classification), and inference (use the weights with approximate sum of products). Several researchers have observed that LNS ALUs in inference may minimize area and power by being both low-precision and approximate (allowing low-cost, tableless implementations). However, the few works that have also trained with LNS report at least part of the system needs accurate LNS. This paper describes a novel approximate LNS ALU implemented simply as logic (without tables) that enables the entire back-propagation training to occur in LNS, at one-third the cost of fixed-point implementation.

Generic programming libraries have historically traded efficiency in return for convenience, and the generics-sop library is no exception. It offers a simple, uniform, representation of all datatypes precisely as a sum of products, making it easy to write generic functions. We show how to finally make generics-sop fast through the use of staging with Typed Template Haskell.

International audience; Dot products (also called sums of products) are ubiquitous in matrix computations, for instance in signal processing. We are especially interested in digital filters, where they are the core operation. We therefore focus on fixed-point arithmetic, used in embedded systems for time and energy efficiency. Common dot product algorithms ensure faithful rounding. For the sake of accuracy and reproducibility, we want to ensure correct rounding. This article describes an algorithm that computes a correctly-rounded sum of products from inputs whose format is known in advance. This algorithm relies on odd rounding (that is easily implemented in hardware) and comes with a careful proof and some cost analysis.

The Newton–Girard Formula allows one to write any elementary symmetric polynomial as a sum of products of power sum symmetric polynomials and elementary symmetric polynomials of lesser degree. It has numerous applications. We have generalized this identity by replacing the elementary symmetric polynomials with monomial symmetric polynomials. Our formula has an application in the field of Lie algebras.

In this work, we present an approach to alleviate the potential benefit of adder graph algorithms by solving the transposed form of the problem and then transposing the solution. The key contribution is a systematic way to obtain the transposed realization with a minimum number of cascaded adders subject to the input realization. In this way, wide and low constant matrix multiplication problems, with sum of products as a special case, which are normally exceptionally time consuming to solve using adder graph algorithms, can be solved by first transposing the matrix and then transposing the solution. Examples show that while the relation between the adder depth of the solution to the transposed problem and the original problem is not straightforward, there are many cases where the reduction in adder cost will more than compensate for the potential increase in adder depth and result in implementations with reduced power consumption compared to using sub-expression sharing algorithms, which can both solve the original problem directly in reasonable time and guarantee a minimum adder depth.

In this paper, we evaluate several families of Toeplitz–Hessenberg determinants whose entries are the Boubaker polynomials. Equivalently, these determinant formulas may be also rewritten as combinatorial identities involving sum of products of Boubaker polynomials and multinomial coefficients. We also present new formulas for Boubaker polynomials via recurrent three-diagonal determinants.

We propose a new technique for the mitigation of fail-stop failures and/or silent data corruptions (SDCs) within linear, sesquilinear or bijective (LSB) operations on $M$ integer data streams ( $M\geq 3$ ). In the proposed approach, the $M$ input streams are linearly superimposed to form $M$ numerically entangled integer data streams that are stored in-place of the original inputs, i.e., no additional (aka. “checksum”) streams are used. An arbitrary number of LSB operations can then be performed in $M$ processing cores using these entangled data streams. The output results can be extracted from any $M-K$ entangled output streams by additions and arithmetic shifts, thereby mitigating $K$ fail-stop failures ( $K\leq \left\lfloor \frac{M-1}{2}\right\rfloor$ ), or detecting up to $K$ SDCs per $M$ -tuple of outputs at corresponding in-stream locations. Therefore, unlike other methods, the number of operations required for the entanglement, extraction and recovery of the results is linearly related to the number of the inputs and does not depend on the complexity of the performed LSB operations. Our proposal is validated within an Amazon EC2 instance (Haswell architecture with AVX2 support) via integer matrix product operations. Our analysis and experiments for fail-stop failure mitigation and SDC detection reveal that the proposed approach incurs 0.75 to 37.23 percent reduction in processing throughput in comparison to the equivalent error-intolerant processing. This overhead is found to be up to two orders of magnitude smaller than that of the equivalent checksum-based method, with increased gains offered as the complexity of the performed LSB operations is increasing. Therefore, our proposal can be used in distributed systems, unreliable multicore clusters and safety-critical applications, where robustness against failures and SDCs is a necessity.

Approximate circuits provide high performance and require low power. Sum-of-products (SOP) units are key elements in many digital signal processing applications. In this brief, three approximate SOP (ASOP) models which are based on the distributed arithmetic are proposed. They are designed for different levels of accuracy. First model of ASOP achieves an improvement up to 64% on area and 70% on power, when compared with conventional unit. Other two models provide an improvement of 32% and 48% on area and 54% and 58% on power, respectively, with a reduced error rate compared with the first model. Third model achieves the mean relative error and normalized error distance as low as 0.05% and 0.009%, respectively. Performance of approximate units is evaluated with a noisy image smoothing application, where the proposed models are capable of achieving higher peak signal-to-noise ratio than the existing state-of-the-art techniques. It is shown that the proposed approximate models achieve higher processing accuracy than existing works but with significant improvements in power and performance.

This work is one of many that are devoted to the further investigation of local interpolating polynomial splines of the fifth order approximation. Here, new polynomial and trigonometrical basic splines are presented. The main features of these splines are the following: the approximation is constructed separately for each grid interval (or elementary rectangular), the approximation constructed as the sum of products of the basic splines and the values of function in nodes and/or the values of its derivatives and/or the values of integrals of this function over subintervals. Basic splines are determined by using a solving system of equations which are provided by the set of functions. It is known that when integrals of the function over the intervals is equal to the integrals of the approximation of the function over the intervals then the approximation has some physical parallel. The splines which are constructed here satisfy the property of the fifth order approximation. Here, the one-dimensional polynomial and trigonometrical basic splines of the fifth order approximation are constructed when the values of the function are known in each point of interpolation. For the construction of the spline, we use the discrete analogues of the first derivative and quadrature with the appropriate order of approximation. We compare the properties of these splines with splines which are constructed when the values of the first derivative of the function are known in each point of interpolation and the values of integral over each grid interval are given. The one-dimensional case can be extended to multiple dimensions through the use of tensor product spline constructs. Numerical examples are represented.

A new method for fast evaluation of high dimensional integrals arising in quantum mechanics is proposed. The method is based on sparse approximation of a high dimensional function followed by a low-rank compression. In the first step, we interpret the high dimensional integrand as a tensor in a suitable tensor product space and determine its entries by a compressed sensing based algorithm using only a few function evaluations. Secondly, we implement a rank reduction strategy to compress this tensor in a suitable low-rank tensor format using standard tensor compression tools. This allows representing a high dimensional integrand function as a small sum of products of low dimensional functions. Finally, a low dimensional Gauss–Hermite quadrature rule is used to integrate this low-rank representation, thus alleviating the curse of dimensionality . Numerical tests on synthetic functions, as well as on energy correction integrals for water and formaldehyde molecules demonstrate the efficiency of this method using very few function evaluations as compared to other integration strategies.

We use shuffle and stuffle relations to give a simple proof of a reduction formula for length-two multiple polylogarithms evaluated in complex parameters of absolute value 1 in terms of a finite sum of products of lengthone polylogarithms. This result was originally due to Nakamura and recently reproved by Panzer by different methods. This generalises results of Borwein and Girgensohn for alternating Euler sums and for multiple zeta values twisted by fourth roots of unity by the first author. We also explore implications for other colored multiple zeta values and present some applications to Mahler measure and Feynman diagrams.

For local homing navigation, a mobile robot is supposed to return home using snapshots of the surrounding environment. It basically follows the snapshot model, comparing the home snapshot and the current view to determine the homing direction. In this paper, we suggest a high-order moment potential to describe the landmark feature distribution for local homing navigation. The moment potential function calculates the sum of products of the feature and the distance of landmark particles as a holistic view, allowing a high order of the distance. It effectively combines the range sensor values of landmarks in the current view and the visual features. By analogy with the moment in physics, the center of the moment is estimated as the reference point, which is the unique convergence point in the convex moment potential from any view, and using the property, the gradient of moment potential at the current position and home location can derive the homing vector. We provide a proof of convergence for any moment potential with order greater than or equal to one. Also, we demonstrate homing performances with various moment models in real environments to validate our models. The suggested moment models combining both landmark distance and visual feature have better performances than the visual information alone, and high-order moment potentials can be searched to obtain a better description of landmark distribution for a given environment.

Consider the Diophantine equation yn=x+x(x+1)+⋯+x(x+1)⋯(x+k), where x, y, n, and k are integers. In 2016, a research article, entitled – ’power values of sums of products of consecutive integers’, primarily proved the inequality n= 19,736 to obtain all solutions (x,y,n) of the equation for the fixed positive integers k≤10. In this paper, we improve the bound as n≤ 10,000 for the same case k≤10, and for any fixed general positive integer k, we give an upper bound depending only on k for n.

The program presented in this paper aims to simplify formulas involving products of a large amount of Wigner 3 j -symbols summed over various magnetic quantum numbers. The algorithm used in our program is based on the graphical techniques originally developed by Yutsis, Levinson, and Vanagas, and later revised by some others. The output of our program is expressed as a weighted sum of products of 3 j -, 6 j - and 9 j -symbols, with appropriate phase factors embedded. Compared with the previous literature, the present program is more flexible in a sense that it can deal with not only closed diagrams, such as the angular momentum recoupling coefficients, but also open diagrams. Our program is particularly useful to high precision atomic and molecular structure calculations using correlated electron–electron coordinates. Program summary Program Title: Reduce3j CPC Library link to program files: https://doi.org/10.17632/9zmx4b5sdd.1 Licensing provisions: BSD 3-clause Programming language: Python 3.6.5 Nature of problem: In atomic and molecular structure calculations, one often encounters expressions containing products of a large amount of Wigner 3 j -symbols summed over various magnetic quantum numbers. It is desirable to simplify these expressions to make the final results compact, which is the goal of this paper. Solution method: The program uses the graphical techniques of quantum mechanical angular momentum theory, originally developed by Yutsis, Levinson, and Vanagas, and later modified by Brink and Satchler. The program is implemented in Python language, with its package “Networkx” extensively used. A friendly graphical input and output interface is developed. The input can be either a recoupling coefficient or an arbitrary expression containing products of the 3 j -symbols summed over various magnetic quantum numbers. A strategically designed optimization method is also developed to help achieve our goal. Our program has significantly expanded the scope of usefulness of the existing programs so that it can now meet most of the demands from high precision atomic and molecular structure calculations. Additional comments including restrictions and unusual features: Reduce3j can effectively handle any open or closed graph, as long as the graph has more than one free line, and all the magnetic angular momentum quantum numbers from internal lines are summed over.

Previously, the synthesis of reversible and quantum permutative circuits based on products of Exclusive-OR (EXOR) sums of inputs (POE) was explored as an approach to reduce quantum cost. Herein, the mathematical representation used in POE expressions is formally detailed and contrasted with the pseudoproduct-based theory of Luccio and Pagli. The theory and practice of product transformation, or computation of equivalent POE expressions, is presented. To demonstrate the generality of the mathematical representations outside of the reversible domain, a new method to synthesize incompletely specified functions as EXOR-AND-OR circuits is introduced. The method is heuristic. It first uses spectral information from the Walsh–Hadamard transform to compute a set of POE implicants for a function, and then uses product transformation to minimize both the total number of literals and unique EXOR sums (ignoring polarity) used in the POE implicants. The product transformation minimization extends to multioutput functions. The new method has no performance guarantees, i.e., it does not have a fallback mode of minimum or near-minimum sum of products; therefore, it is suited to a tournament synthesis approach. When synthesizing benchmark functions of up to 26 inputs, the new method is shown to typically produce lower cost circuits than heuristic pseudoproduct-based methods.

For a (finite or infinite dimensional) vector space V, the notion of a symmetric Jordan canonical form of an operator T ∈ L ( V ) having a minimal polynomial is defined and used to verify the relation between the notions of “Jordan canonical form” and “rational canonical form.” The paper extends and repairs Theorem 2.2 of Radjabalipour (2013) [6] . In particular, it is shown that there exists an auxiliary nilpotent operator S ∈ L ( W ) , depending on T, such that every Jordan canonical form of S yields a symmetric Jordan canonical form and, if the characteristic of the ground field is zero, a rational canonical form for T. The paper concludes with a direct proof of the symmetric Jordan canonical form which “integrates” into a rational canonical form.

A comprehensive treatment on compact representations for the solutions of linear difference equations with variable coefficients, of both n th and unbounded order, is presented. The equivalence between their celebrated combinatorial and determinantal representations is considered. A corresponding representation is proposed using determined nested sums of their variable coefficients. It makes explicit all the sum of products involved in the previous representations of such solutions. Some basic applications are also illustrated.

An exclusive-OR sum of pseudoproducts (ESPP), or a pseudopolynomial over a finite field is a sum of products of linear functions. The length of an ESPP is defined as the number of its pairwise distinct summands. The length of a function f over this field in the class of ESPPs is the minimum length of an ESPP representing this function. The Shannon length function L ESPP (n) on the set of functions over a finite field of k elements in the class of ESPPs is considered; it is defined as the maximum length of a function of n variables over this field in the class of ESPPs. It is proved that L ESPP (n) = O(k n /n 2).

A new method is proposed for a fast evaluation of high-dimensional integrals of potential energy surfaces (PES) that arise in many areas of quantum dynamics. It decomposes a PES into a canonical low-rank tensor format, reducing its integral into a relatively short sum of products of low-dimensional integrals. The decomposition is achieved by the alternating least squares (ALS) algorithm, requiring only a small number of single-point energy evaluations. Therefore, it eradicates a force-constant evaluation as the hotspot of many quantum dynamics simulations and also possibly lifts the curse of dimensionality. This general method is applied to the anharmonic vibrational zero-point and transition energy calculations of molecules using the second-order diagrammatic vibrational many-body Green's function (XVH2) theory with a harmonic-approximation reference. In this application, high dimensional PES and Green's functions are both subjected to a low-rank decomposition. Evaluating the molecular integrals ...

This paper demonstrates both theoretically, through theorem proofs, and experimentally, that existing minimization algorithms for ESOP (Exclusive or Sum Of Products) expressions can be successfully used for minimizing ESCT (Exclusive or Sum of Complex Terms) expressions as well. ESCT expressions are more general than ESOP ones and being reversible are very useful for logic design and for quantum circuits. The field of ESOP minimization has received considerable attention along the years and a great number of research works can be found in the literature, in contrast to the ESCT minimization problem with much fewer works addressing it. The open problems of finding exact ESCT expressions for completely specified functions of more than six variables as well as the problem of finding exact ESCT expressions for incompletely specified functions are solved here based on ESOP principles, for the first time. HighlightsComplex term is a cascade of two variable Boolean functions cells.Complex Terms (CTs) is a generalization of Product Terms (PTs).ESCT and ESOP are exclusive-or sums of (CTs) and (PTs) respectively.It is proved that algorithms for ESOPs can be easily transformed to minimize ESCTs.This transformation is used to solve open problems in ESCT minimization.

In this paper we demonstrate that when one prunes a product basis for CH3CN using g 1 ( n 1 ) + g 2 ( n 2 ) + ⋯ + g D ( n D ) ⩽ b and appropriate g c ( n c ) functions it is possible obtain many converged energy levels using a basis that is smaller than the basis obtained with the pruning condition α 1 n 1 + α 2 n 2 + ⋯ + α D n D ⩽ b (Avila and Carrington, 2011) and comparable to the basis built using residual vectors (Garnier et al., 2016). Importantly, the pruning condition g 1 ( n 1 ) + g 2 ( n 2 ) + ⋯ + g D ( n D ) ⩽ b obviates the need to store a Hamiltonian matrix and facilitates the use of iterative eigensolvers because it is compatible with efficient matrix–vector products. Moreover, it may be used in conjunction with a nondirect product quadrature and therefore allows one to use a potential energy surface that is not a simple sum of products.

Spiking convolutional neural networks have become a novel approach for machine vision tasks, due to the latency to process an input stimulus from a scene, and the low power consumption of these kind of solutions. Event-based systems only perform sum operations instead of sum of products of framebased systems. In this work an upgrade of a neuromorphic event-based convolution accelerator for SCNN, which is able to perform multiple layers with different kernel sizes, is presented. The system has a latency per layer from 1.44 μs to 9.98μs for kernel sizes from 1x1 to 7x7.

This paper deals with approximate Sum-of-Products (SOP) models and its application in linear convolution. SOP unit finds its application in many DSP applications like convolution and design of filters. In this brief, three Approximate Sum-of-Products (ASOP) models have achieved comparable improvements in area and power. SOP units use the concept of distributed arithmetic to find the vector products without the use of multipliers. Performance is justified by using the distributed arithmetic model for linear convolution of input vectors. The approximate models have achieved significant improvements in area power parameters.

Standard multiconfiguration time-dependent Hartree (MCTDH) calculations use a direct product basis and rely on the potential being a sum of products (SOPs). The size of the direct product MCTDH basis scales exponentially with the number of atoms. Accurate potentials may not be SOPs. We introduce an MCTDH approach that uses a pruned basis and a collocation grid. Pruning the basis significantly reduces its size. Collocation makes it possible to do calculations using a potential that is not a SOP. The collocation point set is a Smolyak grid. Strategies using pruned MCTDH bases already exist, but they work only if the potential is a SOP. Strategies for using MCTDH with collocation also exist, but they work only if the MCTDH basis is a direct product. In this paper, we combine a pruned basis with collocation. This makes it possible to mitigate the direct-product basis size problem and do calculations when the potential is not a SOP. Because collocation is used, there are no integrals and no need for quadrature. All required matrix-vector products can be evaluated sequentially. We use nested sets of collocation points and hierarchical basis functions. They permit efficient inversion of the (large) matrix whose elements are basis functions evaluated at points, which is necessary to transform values of functions at points to basis coefficients. The inversion technique could be used outside of chemical physics. We confirm the validity of this new pruned, collocation-based (PC-)MCTDH approach by calculating the first 50 vibrational eigenenergies of CH2NH.

In this paper we investigate when a finite sum of products of two Toeplitz operators with quasihomogeneous symbols is a finite rank perturbation of another Toeplitz operator on the Bergman space. We discover a noncommutative convolution ⋄ on the space of quasihomogeneous functions and use it in solving the problem. Our main results show that if F j , G j ( 1 ≤ j ≤ N ) are polynomials of z and z ¯ then ∑ j = 1 N T F j T G j − T H is a finite rank operator for some L 1 -function H if and only if ∑ j = 1 N F j ⋄ G j belongs to L 1 and H = ∑ j = 1 N F j ⋄ G j . In the case F j 's are holomorphic and G j 's are conjugate holomorphic, it is shown that H is a solution to a system of first order partial differential equations with a constraint.

In this paper, we describe the results of the experimental comparison of programs that implement various decomposition methods for disjunctive normal forms of systems of completely defined Boolean functions. The complexity of a system of disjunctive normal forms is expressed in two ways: by the area of a programmable logic array that implements a system of disjunctive normal forms, or by the number of vertices of a binary decision diagram, which represents a system of Boolean functions. The complexity of the functional expansion of a system's functions is determined as the sum of the complexities of the subsystem of the functions included in this expansion. The estimates of the complexity are oriented on the synthesis of combinational circuits based on the programmed logical arrays and the library's logical elements.

We show that the Kronecker canonical form (which is a canonical decomposition for pairs of matrices) is the representation of a linear relation in a finite dimensional space. This provides a new geometric view upon the Kronecker canonical form. Each of the four entries of the Kronecker canonical form has a natural meaning for the linear relation which it represents. These four entries represent the Jordan chains at finite eigenvalues, the Jordan chains at infinity, the so-called singular chains and the multi-shift part. Or, to state it more concisely: For linear relations the Kronecker canonical form is the analogue of the Jordan canonical form for matrices.

In this work, an approach for transposing solutions to the multiple constant multiplication (MCM) problem to obtain a sum of product (SOP) computation with minimum depth is proposed. The reason for doing this is that solving the SOP problem directly is highly computationally intensive when adder graph algorithms are used. Compared to using sub-expression sharing algorithms, which has a lower computational complexity, directly for the SOP problem, it is shown that the proposed approach, as expected, results in lower complexity for the SOP. It is also shown that there is no obvious way to construct the MCM solution in such a way that the SOP solution has the minimum theoretical depth. However, the proposed approach guarantees minimum depth subject to the MCM solution given as input.

We discuss the exact plaquette-ordered ground states of the generalized Hubbard model on the Kagom\'e lattice for several fillings, by constructing the Hamiltonian as a sum of products of projection operators for up and down spin sectors. The obtained exact ground states are interpreted as N\'eel ordered states on the bond-located electrons. We determine several parameter regions of the exact ground states, and calculate the entanglement entropy. We examine the above results by numerical calculations based on exact diagonalization and density-matrix renormalization group methods., Comment: 11 pages, 13 figures. to appear in EPJ B topical issue "Coexistence of long-range orders in low-dimensional systems"