Unique Decomposition Research Articles

Maximum bonding fragment orbitals for deciphering complex chemical interactions.

An optimal set of fragment orbitals is proposed that offers a simple and general way to describe complex bonding situations. The idea is based on the unique decomposition of the total bond order between fragments. Each resulting component corresponds to the bonding between a pair of optimal fragment orbitals. These pairs further form a set of exactly doubly occupied bond orbitals, which fully describes the chemical bonding between two fragments. Thereby, complex bonding interactions can be separated into a few well-defined pairwise orbital interactions. A remarkable feature of the theory is that one can build up a quantitative and clearly correlated orbital interaction diagram. Explicit analytical expressions are also presented for bond orders, bond polarities, occupancies, and orbital interaction energies, providing new insights into the bond order concept and the fundamental principles of molecular orbital theory. Through illustrative examples, this new approach is shown to be a simple and powerful tool for analyzing different kinds of chemical interactions, including covalent, ionic and noncovalent bonding.

Effective identifiability criteria for tensors and polynomials

A tensor T, in a given tensor space, is said to be h-identifiable if it admits a unique decomposition as a sum of h rank one tensors. A criterion for h-identifiability is called effective if it is satisfied in a dense, open subset of the set of rank h tensors. In this paper we give effective h-identifiability criteria for a large class of tensors. We then improve these criteria for some symmetric tensors. For instance, this allows us to give a complete set of effective identifiability criteria for ternary quintic polynomials. Finally, we implement our identifiability algorithms in Macaulay2.

On the number of Waring decompositions for a generic polynomial vector

We prove that a general polynomial vector (f1,f2,f3) in three homogeneous variables of degrees (3,3,4) has a unique Waring decomposition of rank 7. This is the first new case we are aware of, and likely the last one, after five examples known since the 19th century and the binary case. We prove that there are no identifiable cases among pairs (f1,f2) in three homogeneous variables of degree (a,a+1), unless a=2, and we give a lower bound on the number of decompositions. The new example was discovered with Numerical Algebraic Geometry, while its proof needs Nonabelian Apolarity.

Virtually semisimple modules and a generalization of the Wedderburn-Artin theorem

ABSTRACTA widely used result of Wedderburn and Artin states that “every left ideal of a ring R is a direct summand of R if and only if R has a unique decomposition as a finite direct product of matrix rings over division rings.” Motivated by this, we call a module M virtually semisimple if every submodule of M is isomorphic to a direct summand of M and M is called completely virtually semisimple if every submodule of M is virtually semisimple. We show that the left R-module R is completely virtually semisimple if and only if R has a unique decomposition as a finite direct product of matrix rings over principal left ideal domains. This shows that R is completely virtually semisimple on both sides if and only if every finitely generated (left and right) R-module is a direct sum of a singular module and a projective virtually semisimple module. The Wedderburn-Artin theorem follows as a corollary from our result.

Decompositions in Complete Lattices III. Unique Irredundant Decompositions and Convex Geometries

We give a characterization of complete strongly dually atomic lattices having unique irredundant decompositions which are also canonical. It is shown that all known characterizations of lattices with unique irredundant decompositions are a consequence of this result. In addition, upper continuous closure lattices of convex geometries with (unique) irredundant decompositions are characterized.

Peculiarities of Pyrolysis of Hydrolytic Lignin in Dispersed Gas Phase and in Solid State

The unique decomposition pathways of hydrolytic lignin (HL) dissolved in an acetone/water mixture (9:1) and dispersed by a droplet evaporation technique under nitrogen gas flow has been investigated in a conventional reactor at atmospheric condition, a temperature region of 400–550 °C, and a residence time of 0.12 s. The results validate the fact that dispersion of the lignin into the gas phase by decreasing the sample size (as well as “minimizing the char area to avoid catalytic contact” of molecular products/radicals with the surface) may open new perspectives in understanding the chemistry of the depolymerization of lignin. Using Laser Desorption Ionization-Time of Flight-Mass Spectrometry (LDI-TOF-MS) the intrinsic ion m/z = 550, as the major MS peak from fresh HL dissolved in an acetone/water mixture before pyrolysis, was detected. Surprisingly, the expected phenolic compounds after pyrolysis were in trace amounts at less than 15% conversion of lignin. Instead, oligomeric intermediate substances with...

Canonical decomposition of belief functions based on Teugels’ representation of the multivariate Bernoulli distribution

A canonical decomposition of belief functions is a unique decomposition of belief functions into elementary pieces of evidence. Smets found an equivalent representation of belief functions, which he interpreted as a canonical decomposition. However, his proposal is not entirely satisfactory as it involves elementary pieces of evidence, corresponding to a generalisation of belief function axioms, whose semantics lacks formal justifications. In this paper, a new canonical decomposition relying only on well-defined concepts is proposed. In particular, it is based on a means to induce belief functions from the multivariate Bernoulli distribution and on Teugels’ representation of this distribution, which consists of the means and the central moments of the underlying Bernoulli random variables. According to our decomposition, a belief function results from as many crisp pieces of information as there are elements in its domain, and from simple probabilistic knowledge concerning their marginal reliability and the dependencies between their reliability. In addition, we show that instead of interpreting with some difficulty Smets’ representation of belief functions as a canonical decomposition, it is possible to give it a different and well-defined semantics in terms of measures of information associated with the reliability of the pieces of information in our decomposition.

Changes in vertex status and the fundamental decomposition of a tree relative to a multiple (parter) eigenvalue

Given a tree T, a Hermitian matrix A whose graph is T and an eigenvalue λ of A, a number of new structural results about T relative to the multiplicity of λ in A are developed. These include a complete classification of the possible changes in status of one vertex upon removal of another. These are used, in part, to give a unique fundamental decomposition of the tree that can be used to answer further structural questions. In the process, the notions of singly and multiply Parter vertices are introduced and used. Possible changes in the structure of the fundamental decomposition, resulting from changes in one diagonal entry, or deletion of a row and a column, are also discussed.

The Main Eigenvalues of the Undirected Power Graph of a Group

The undirected power graph of a finite group $G$, $P(G)$, is a graph with the group elements of $G$ as vertices and two vertices are adjacent if and only if one of them is a power of the other. Let $A$ be an adjacency matrix of $P(G)$. An eigenvalue $lambda$ of $A$ is a main eigenvalue if the eigenspace $epsilon(lambda)$ has an eigenvector $X$ such that $X^{t}jjneq 0$, where $jj$ is the all-one vector. In this paper we want to focus on the power graph of the finite cyclic group $mathbb{Z}_{n}$ and find a condition on n where $P(mathbb{Z}_{n})$ has exactly one main eigenvalue. Then we calculate the number of main eigenvalues of $P(mathbb{Z}_{n})$ where $n$ has a unique prime decomposition $n = p^{r} p_2$. We also formulate a conjecture on the number of the main eigenvalues of $P(mathbb{Z}_{n})$ for an arbitrary positive integer $n$.

Projector representation of isotropic linear elastic material laws for directed surfaces

AbstractIn the framework of linear elasticity, it is possible to use eigenspace projectors to describe the elasticity tensor, at least for special cases of material symmetries. A similar procedure is also advantageous in the context of directed surfaces. It is thus possible to introduce this representation to stiffness measures of thin‐walled members. Hereby, we reduce our considerations to an elastic mid‐surface, where the in‐plane, out‐of‐plane, and transverse shear states are uncoupled, but superposed eventually. Thereby, we introduce a unique decomposition of the stiffness measures for the considered deformation states, while we limit our considerations to homogeneous materials without pronounced orientation dependency. For each of these three states, eigenvalues of the stiffness tensors are evaluated based on engineering material parameters. Finally, the whole procedure allows for the clear distinction of dilatoric and deviatoric portions in the constitutive equations.After all, a compact and mathematically easy‐to‐handle representation for the stiffness tensors with respect to in‐plane, out‐of‐plane, and transverse shear state has been found. Thereby, we show correlations to classical representations as well as advantages due to the clarity of present scheme.

Block Decomposition For Modules

Block decomposition for rings has been introduced and shown to be unique in the literature (see [T. Y. Lam, Graduate Texts in Mathematics, 131, Springer-Verlag, New York, 1991]). Applying annihilator submodules, we extend this definition to modules and show that every module $M$ has a unique block decomposition $M=\bigoplus_{i=1}^nM_i$ where each $M_i$ is an annihilator submodule. We also show that the block decomposition for any ring $R$ and the block decomposition for the module $R_R$, are identical. Block decomposition provides us with a decomposition for $\edmp{M}$ because $\edmp{M}\iso\prod_{i=1}^n\edmp{M_i}$.

Real identifiability vs. complex identifiability

A real tensor T of rank r is identifiable if it has a unique decomposition with rank 1 tensors. Sometimes identifiability fails over , for general tensors of fixed rank. This behaviour is peculiar when r is sub-generic. In several cases, the failure is due to an elliptic normal curve passing through general points of the corresponding variety of rank 1 tensors (Segre, Veronese, Grassmann). When this happens, we prove the existence of nonempty euclidean open subsets of some varieties of rank r tensors, whose elements have several complex decompositions, but only one of them is real.

A divisibility result in combinatorics of generalized braids

For every finite Coxeter group Γ, each positive braid in the corresponding braid group admits a unique decomposition as a finite sequence of elements of Γ, the so-called Garside-normal form. The study of the associated adjacency matrix Adj(Γ) allows to count the number of Garside-normal form of a given length. In this paper we prove that the characteristic polynomial of Adj(Bn) divides the one of Adj(Bn+1). The key point is the use of a Hopf algebra based on signed permutations. A similar result was already known for the type A. We observe that this does not hold for type D. The other Coxeter types (I, E, F and H) are also studied.

Treatment of ramie fiber with different techniques: the influence of diammonium phosphate on interfacial adhesion properties of ramie fiber-reinforced polylactic acid composite

Ramie fiber-reinforced polylactic acid (PLA) composites were successfully prepared by hot compression molding. Different treatment techniques were used to modify the surface of ramie fiber. The influence of diammonium phosphate (DAP) on the interfacial adhesion between ramie fiber and PLA composites was investigated by the contact angle measurements, FTIR and SEM analyses. The contact angle measurement results showed that alkali treatment combined with DAP was very efficient in decreasing the hydrophilicity of fibers. After treatment, the hydrophilicity of untreated ramie fiber from 5.9 ± 1.3 decreased to 2.0 ± 0.8 mJ/m2. The wettability of alkali/silane/DAP-treated ramie fiber/PLA composite was higher (95.4° ± 1.3°) than that of pure ramie fiber/PLA composite (87.3° ± 1.9°). The FTIR results were consistent with the wetting measurements as the increment of hydrophilicity. Thermal analysis indicated that DAP-modified ramie fiber/PLA composites exhibited a lower thermal decomposition temperature, unique decomposition behavior and more residual char formation at decomposition temperature. The tensile, flexural and impact properties of DAP-modified ramie fiber composites were comparable to those of untreated ramie fiber composite. Moreover, proper alignment and uniform distribution of ramie fibers within the PLA matrix were found to be excellent. The morphological structures observed by SEM showed that well-modified ramie fibers enhanced the failure of the PLA composites in tensile, flexural and impact tests.

On rotational ambiguity in parallel profiles with linear dependencies

Several curve resolution techniques have been developed to analyze higher‐order data structures, among which one can name parallel profiles with linear dependencies (PARALIND). The PARALIND is a variant of the Parallel Factor Analysis (PARAFAC) to cope with linear dependencies. Applying a PARAFAC model with different random initializations to a noisy three‐way data with linearly dependent loadings always converges to a best‐fitted noisy solution. Since PARAFAC maximally fits the noise part of the data, rank‐deficiency is broken, and it causes a noise‐induced unique decomposition. The uniqueness due to the presence of noise in rank‐deficient cases is an artificial phenomenon called “surface uniqueness.” The PARALIND model causes non‐uniqueness to be revealed, ie, it leads to a different set of profiles in each run. However, it might not give us all the possible solutions. Therefore, there is still a demand for determining all and every possible solution. The aim of this paper is 3‐fold: (1) A method is developed for determining feasible regions using the PARALIND model. (2) The surface uniqueness phenomenon is clarified. (3) The effect of interaction matrices in PARALIND as an active constraint is emphasized and discussed in detail using simulated noisy and noise‐free two‐ and three‐component data arrays.

Degradation of Hole Transport Materials via Exciton-Driven Cyclization

Organic light-emitting diode (OLED) displays have been an active and intense area of research for well over a decade and have now reached commercial success for displays from cell phones to large format televisions. A more thorough understanding of the many different potential degradation modes which cause OLED device failure will be necessary to develop the next generation of OLED materials, improve device lifetime, and to ultimately improve the cost vs performance ratio. Each of the different organic layers in an OLED device can be susceptible to unique decomposition pathways, however stability toward excitons is critical for emissive layer (EML) materials as well as any layer near the recombination zone. This study will specifically focus on degradation modes within the hole transport layer (HTL) with the goal being to identify the general decomposition paths occurring in an operating device and use this information to design new derivatives which can block these pathways. Through post-mortem analyses of several aged OLED devices, an apparently common intramolecular cyclization pathway has been identified that was not previously reported for arylamine-containing HTL materials and that operates parallel to but faster than the previously described fragmentation pathways.

Fluorescence Approach for the Determination of Fluorescent Dissolved Organic Matter.

Excitation-emission matrix (EEM) fluorescence spectroscopy coupled with parallel factor (PARAFAC) analysis has been widely applied to characterize dissolved organic matter (DOM) in aquatic and terrestrial systems. However, its application in environmental samples is limited because PARAFAC is not able to handle nontrilinear EEM data, leading to the overestimated number of components and incorrect decomposition results. In this work, a new method, parallel factor framework-clustering analysis (PFFCA), is proposed to resolve this problem. First, simulated data with different signal-to-noise ratios and intensities of nontrilinear structure were tested to confirm the robustness of PFFCA. The residual sum of squares (RSS) of PARAFAC was significantly higher than that of PFFCA (p < 0.037). Second, a set of data originating from a synthetic mixture of humic acid and bovine serum albumin was applied to compare with PARAFAC with known samples. PFFCA provided an estimation (R2 > 0.92) closer to actual EEM than PARAFAC (R2 > 0.81). Finally, to confirm the feasibility of PFFCA in analyzing natural samples, DOM-containing samples collected from both a polluted lake and river were tested, indicating that PFFCA provides a more precise estimation than PARAFAC. The results clearly indicate that PFFCA offers a robust approach for the unique decomposition of complex synthetic and natural samples, which is of great significance in understanding the characteristics of DOM in aqueous systems.

Synergy and Redundancy in Dual Decompositions of Mutual Information Gain and Information Loss

Williams and Beer (2010) proposed a nonnegative mutual information decomposition, based on the construction of information gain lattices, which allows separating the information that a set of variables contains about another variable into components, interpretable as the unique information of one variable, or redundant and synergy components. In this work, we extend this framework focusing on the lattices that underpin the decomposition. We generalize the type of constructible lattices and examine the relations between different lattices, for example, relating bivariate and trivariate decompositions. We point out that, in information gain lattices, redundancy components are invariant across decompositions, but unique and synergy components are decomposition-dependent. Exploiting the connection between different lattices, we propose a procedure to construct, in the general multivariate case, information gain decompositions from measures of synergy or unique information. We then introduce an alternative type of lattices, information loss lattices, with the role and invariance properties of redundancy and synergy components reversed with respect to gain lattices, and which provide an alternative procedure to build multivariate decompositions. We finally show how information gain and information loss dual lattices lead to a self-consistent unique decomposition, which allows a deeper understanding of the origin and meaning of synergy and redundancy.

Unique Decomposition and a New Model for the Ground Moving Target Indication Problem

In wide-area surveillance radar systems, ground moving target indication is the main task. The underlying mathematical problem is to decompose a complex matrix into a low rank matrix and a structured sparse matrix. In this paper, we show that such decomposition has a unique solution under reasonable assumptions. We propose a phase-based model to fully describe the special sparse structure. An alternating direction method of multipliers is implemented to solve the resulting nonconvex complex matrix problem. Simulation results verify the superior efficiency and the improvement of the new model.

The Iterative Nature of the “Zig” and How to Define the “Hows”

The AD's design equation depicts the relationship between the functional requirements (FR) and the design parameters (DP) by the design matrix (DM), through a unique zigzag decomposition path. At the “zig part” of each level of the zigzag decomposition, the designer needs to find out the DPs that can fulfil the given FRs. This paper proposes that the designer has to perform three main actions in a zig process in order to define the design equation: to define the DPs at a nominal condition and its magnitude; to evaluate the interactions of the DP with the system at actual conditions; and to check back the set of FRs verifying if they fit inside the design range. The purpose of this paper is to illustrate the actions performed on a zig, emphasising the changes that may occur in the arrangement of the design during the synthesis of the DM at any level of the decomposition. At each level of decomposition, the estimation of the DPs that fulfil the FRs allows the designer to define a subset of the DM, making it possible to evaluate afterwards the DM with all the interactions of the system. Moreover, in what concerns to the information content, it is possible to evaluate the probability of success of the system taking into account the interactions of the system and the tolerances of the DPs. This paper presents an example regarding the evaluation of the DM using the equations of the design for a variable air volume (VAV) air conditioning system.