Search

Part I. Presents information on a study which determined the average errors capacity region of an arbitrarily varying multiple-access channel (AVMAC) using elimination technique. Description of a decoding rule for AVMAC; Deviational properties.

Let f:M’N be a stable map between orientable 4-manifolds, where M is closed and N is stably parallelisable. It is shown that the signature of M vanishes if and only if there exists a stable map g:M’N homotopic to f which has only fold and cusp singularities. This together with results of Ando and Èliaaberg shows that, in this situation, the Thom polynomials are the only obstructions to the elimination of the singularities except for the fold singularity. Also studied are some topological properties (including those of the discriminant set) of stable maps between 4-manifolds with only Ak-type singularities. [ABSTRACT FROM AUTHOR]

The paper introduces NJg, a variant of Gentzen's NJ natural deduction system, in which the implication elimination rule has no minor premise. The NJg-systems extends traditional ND-systems with a new kind of action in derivations, assumption incorporation, a kind of dual to the assumption discharge action. As a result, the implication (I/E)-rules are invertible and, almost by definition, harmonious and stable, a major condition imposed by proof-theoretic semantics on ND-systems to qual-ify as meaning-conferring. There is also a proof-term assignment to NJg- derivations, materialising the Curry-Howard correspondence for this system. [ABSTRACT FROM AUTHOR]

Let K be a field, and a finitely generated K-algebra with the PBW K-basis . It is shown that if L is a nonzero left ideal of A with GK.dim(A∕L) = d<n ( =  the number of generators of A), then L has the elimination property in the sense that V(U)∩L≠{0} for every subset with , where V(U) = K-span. In terms of the structural properties of A, it is also explored when the condition GK.dim(A∕L)<n may hold for a left ideal L of A. Moreover, from the viewpoint of realizing the elimination property by means of Gröbner bases, it is demonstrated that if A is in the class of binomial skew polynomial rings or in the class of solvable polynomial algebras, then every nonzero left ideal L of A satisfies GK.dim(A∕L)< GK.dimA = n ( =  the number of generators of A), thereby L has the elimination property. [ABSTRACT FROM AUTHOR]

Generalizing Cooper’s method of quantifier elimination for Presburger arithmetic, we give a new proof that all parametric Presburger families {St:t∈N} [as defined by Woods (Electron J Comb 21:P21, 2014)] are definable by formulas with polynomially bounded quantifiers in an expanded language with predicates for divisibility by f(t) for every polynomial f∈Z[t]. In fact, this quantifier bounding method works more generally in expansions of Presburger arithmetic by multiplication by scalars {α(t):α∈R,t∈X} where R is any ring of functions from X into Z. [ABSTRACT FROM AUTHOR]

We investigate cut elimination in multi-focused sequent calculi and the impact on the cut elimination proof of design choices in such calculi. The particular design we advocate is illustrated by a multi-focused calculus for full linear logic using an explicitly polarised syntax and incremental focus handling, for which we provide a syntactic cut elimination procedure. We discuss the effect of cut elimination on the structure of proofs, leading to a conceptually simple proof exploiting the strong structure of multi-focused proofs. [ABSTRACT FROM AUTHOR]

The QSS codes associated with a MSP code are based on finding an invertible matrix V, solving the system vATMBsa=s. We propose a quantum Gauss-Jordan Elimination Procedure to produce such a pivotal matrix V by using the Grover search code. The complexity of solving is of square-root order of the cardinal number of the unauthorized set 2|B|. [ABSTRACT FROM AUTHOR]

In this paper, an extension Elimination and Choice Translating Reality (ELECTRE) method is introduced to handle multi-valued neutrosophic multi-criteria decision-making (MCDM) problems. First of all, some outranking relations for multi-valued neutrosophic numbers (MVNNs), which are based on traditional ELECTRE methods, are defined, and several properties are analyzed. In the next place, an outranking method to deal with MCDM problems similar to ELECTRE III, where weights and data are in the form of MVNNs, is developed. At last, an example is provided to demonstrate the proposed approach and testify its validity and feasibility. This study is supported by the comparison analysis with other existing methods. [ABSTRACT FROM AUTHOR]

The relationship among first-order monadic omega-logic (MOL), propositional (until-free) linear-time temporal logic (LTL), propositional generalized definitional reflection logic (GDRL) and propositional infinitary logic (IL) is clarified via embedding theorems. A theorem for embedding a Gentzen-type sequent calculus MOω for MOL into a Gentzen-type sequent calculus LTω for LTL is proved. The cut-elimination theorem for MOω is proved using this embedding theorem. MOL is also shown to be decidable through the use of this embedding theorem. Theorems for embedding LTω into MOω and MOω into a Gentzen-type sequent calculus LKω for IL are also proved. Moreover, a theorem for embedding MOω into a Gentzen-type sequent calculus GDω for GDRL and a theorem for embedding LTω into GDω are proved. [ABSTRACT FROM AUTHOR]

Excised etiolated cotyledons of cucumber (Cucumis sativus L. cv. Aonagajibai) were continuously irradiated under various intensities of white light. The rate of chlorophyII (Chi) formation during the lag phase reaches a plateau at fluence rates above 1.4 urmol m−2 s−1. This is true in both water‐control and benzyladenine (BA)‐pretreated cotyledons. In cotyledons pretreated for 14 h with BA in darkness (in which case, Chl formation is stimulated by BA during both the lag and the steady‐state phases), the increase in the steady‐state rate of Chl formation with increasing light in tensity is stimulated compared to that of the water control over the range of fluence rates, 0. 25‐43 urmol m−2 s−1. In cotyledons pretreated for 6 h with BA in darkness (only Chl formation during the lag phase is stimulated), only an increase in fluence rate from 0.25 to 1.4 umol m−2 s−1 causes a higher increase in the Chl formation in the BA‐treated cotyledons than in the water control. The time course of Chl formation shows that the BA‐induced late‐appearing effect (stimulation of the steady‐state rate) is almost absent at low intensity illumination, but the BA‐induced fast‐appearing effect (elimination of the lag phase) is effective at all intensities. From this evidence, the Chl‐forming process apparently consists of two components, whose periods of operation or light‐intensity requirements are different. BA stimulates the rates of the respective components in both the fast and the late‐appearing effects. [ABSTRACT FROM AUTHOR]

Focuses on the response elimination procedure. Information on experiments which were conducted; Information on methodologies used; Results and discussions of the experiments and methodologies; Indepth look at the elimination procedure.

Eliminating nuisance parameters from a model is universally recognized as a major problem of statistics. A surprisingly large number of elimination methods have been proposed by various writers on the topic. In this article we propose to critically review two' such elimination methods. We shall be concerned with some particular cases of the marginalizing and the conditioning methods. The origin of these methods may be traced to the work of Sir Ronald A. Fisher. The contents of the marginalization and the conditionality arguments are then reexamined from the Bayesian point of view. This article should be regarded as a sequel to the author's three-part essay (Basu 1975) on statistical information and likelihood. [ABSTRACT FROM AUTHOR]

The article describes a modification of LD and proves the cut-elimination theorem for it. It establishes a weak version of cut-elimination theorem for LD, stating that all cuts except some special forms can be eliminated from a proof in LD. It obtains some corollaries on syntactical properties of CD from these cut-elimination theorems.

Discusses the application of a cut-elimination theorem of mathematician Gerhard Gentzen in the alpha graphs logic of philosopher Charles Peirce. Infinite supply of atomic formulas; Even elimination and odd introduction as rules of inference; Overview of the rule of insertion and deletion.

We introduce a new family of sequential selection procedures wherein the subsets selected have random sizes. In comparison to subset selection procedures that select subsets of fixed size, the new procedures alleviate the need to specify the subset size prior to the experiment. We discuss the application of such procedures in the context of early phase clinical trials. The new procedures retain the adaptive features of the Levin-Robbins-Leu family of sequential subset selection procedures for selecting subsets of fixed size, namely, sequential elimination of inferior treatments and sequential recruitment of superior treatments. These two adaptive features respectively address ethical concerns that diminish interest in nonadaptive procedures and also allow promising treatments to be brought forward for further testing without having to wait until the end of the trial. The new procedures differ from the classical subset selection procedures of Shanti S. Gupta in terms of their respective goals and operating characteristics and we compare the two approaches in a simulation study. The findings suggest that whereas Gupta’s procedure achieves its goal of including the single best treatment in the final selected subset with high probability, it does so by virtue of a nonadaptive, fixed sample size procedure that lacks necessary flexibility in the context of clinical research. By contrast, the new procedures aim to select treatment subsets that satisfy a different criterion, that ofacceptable subset selectionwith high probability, while allowing adaptive elimination and recruitment and other flexibilities which we discuss to fit the practical needs of selection methods in clinical research. [ABSTRACT FROM PUBLISHER]

It is known that the β-conversions of the full intuitionistic propositional calculus (IPC) translate into βη-conversions of the atomic polymorphic calculus Fat . Since Fat enjoys the property of strong normalization for βη-conversions, an alternative proof of strong normalization for IPC considering β-conversions can be derived. In the present article, we improve the previous result by analysing the translation of the η-conversions of the latter calculus into a technical variant of the former system (the atomic polymorphic calculus F∧at ). In fact, from the strong normalization of F∧at we can derive the strong normalization of the full intuitionistic propositional calculus considering all the standard (β and η) conversions. [ABSTRACT FROM AUTHOR]

Algebraic elimination of nodes at discrete fracture intersections via the star-delta technique has proven to be a valuable tool for making multiphase numerical simulations more tractable and efficient. This study examines the assumptions of the star-delta technique and exposes its effects in a 3-D, multiphase context for advective and dispersive/diffusive fluxes. Key issues of relative permeability-saturation-capillary pressure ( kr-S-Pc) and capillary barriers at fracture-fracture intersections are discussed. This study uses a multiphase compositional, finite difference numerical model in discrete fracture network (DFN) and discrete fracture-matrix (DFM) modes. It verifies that the numerical model replicates analytical solutions and performs adequately in convergence exercises (conservative and decaying tracer, one and two-phase flow, DFM and DFN domains). The study culminates in simulations of a two-phase laboratory experiment in which a fluid invades a simple fracture intersection. The experiment and simulations evoke different invading fluid flow paths by varying fracture apertures as oil invades water-filled fractures and as water invades air-filled fractures. Results indicate that the node elimination technique as implemented in numerical model correctly reproduces the long-term flow path of the invading fluid, but that short-term temporal effects of the capillary traps and barriers arising from the intersection node are lost. [ABSTRACT FROM AUTHOR]

Fourier-Motzkin elimination is a computationally expensive but powerful method to solve a system of linear inequalities. These systems arise e.g. in execution order analysis for loop nests or in integer linear programming. This paper focuses on the analysis, design and implementation of a parallel solver for distributed memory for large systems of linear inequalities using the Fourier-Motzkin elimination algorithm. We also measure the speedup of parallel solver and prove that this implementation results in good scalability. [ABSTRACT FROM AUTHOR]

We study a class of homotopies between maps from 4-manifolds to surfaces which we call cusp merges. These homotopies naturally appear in the uniqueness problems for certain pictorial descriptions of 4-manifolds derived from maps to the 2-sphere (for example, broken Lefschetz fibrations, wrinkled fibrations, or Morse 2-functions). Our main results provide a classification of cusp merge homotopies in terms of suitably framed curves in the source manifold, as well as a fairly explicit description of a parallel transport diffeomorphism associated to a cusp merge homotopy. The latter is the key ingredient in understanding how the aforementioned pictorial descriptions change under homotopies involving cusp merges. We apply our methods to the uniqueness problem of surface diagrams of 4-manifolds and describe algorithms to obtain surface diagrams for total spaces of (achiral) Lefschetz fibrations and 4-manifolds of the form M × S1, where M is a 3-manifold. Along the way we provide extensive background material about maps to surfaces and homotopies thereof and develop a theory of parallel transport that generalizes the use of gradient flows in Morse theory [ABSTRACT FROM AUTHOR]

Abstract: α-Mesyloxyoxetanes undergo stereospecific elimination to 2-alkylidene oxetanes upon treatment with potassium t-butoxide. [Copyright &y& Elsevier]

Inferentialism claims that the rules for the use of an expression express its meaning without any need to invoke meanings or denotations for them. Logical inferentialism endorses inferentialism specifically for the logical constants. Harmonic inferentialism, as the term is introduced here, usually but not necessarily a subbranch of logical inferentialism, follows Gentzen in proposing that it is the introduction-rules which give expressions their meaning and the elimination-rules should accord harmoniously with the meaning so given. It is proposed here that the logical expressions are those which can be given schematic rules that lie in a specific sort of harmony, general-elimination (ge) harmony, resulting from applying a certain operation, the ge-procedure, to produce ge-rules in accord with the meaning defined by the I-rules. Griffiths (2014) claims that identity cannot be given such rules, concluding that logical inferentialists are committed to ruling identity a nonlogical expression. It is shown that the schematic rules for identity given in Read (2004), slightly amended, are indeed ge-harmonious, so confirming that identity is a logical notion. [ABSTRACT FROM AUTHOR]

The $$\Omega $$ -rule was introduced by W. Buchholz to give an ordinal-free proof of cut-elimination for a subsystem of analysis with $$\Pi ^{1}_{1}$$ -comprehension. W. Buchholz's proof provides cut-free derivations by familiar rules only for arithmetical sequents. When second-order quantifiers are present, they are introduced by the $$\Omega $$ -rule and some residual cuts are not eliminated. In the present paper, we introduce an extension of the $$\Omega $$ -rule and prove the complete cut-elimination by the same method as W. Buchholz: any derivation of arbitrary sequent is transformed into its cut-free derivation by the standard rules (with induction replaced by the $$\omega $$ -rule). In fact we treat the subsystem of $$\Pi ^{1}_{1}$$ -CA (of the same strength as $$ID_{1}$$ ) that W. Buchholz used for his explanation of G. Takeuti's finite reductions. Extension to full $$\Pi ^{1}_{1}$$ -CA is planned for another paper. [ABSTRACT FROM AUTHOR]

Motivated by a recent Basu–Martin–Ryan paper, we obtain a reduced primal-dual pair of a linear semi-infinite programming problem by applying an amended Fourier–Motzkin elimination method to the linear semi-infinite inequality system. The reduced primal-dual pair is equivalent to the original one in terms of consistency, optimal values and asymptotic consistency. Working with this reduced pair and reformulating a linear semi-infinite programme as a linear programme over a convex cone, we reproduce all the theorems that lead to the full eleven possible duality state classification theory. Establishing classification results with the Fourier–Motzkin method means that the two classification theorems for linear semi-infinite programming, 1969 and 1974, have been proved by new and exciting methods. We also show in this paper that the approach to study linear semi-infinite programming using Fourier–Motzkin elimination is not purely algebraic, it is mixed algebraic-analysis. [ABSTRACT FROM AUTHOR]

This study compared four lineup procedures: the simultaneous, sequential, elimination, and wildcard. Two hundred and sixty-nine university students ( M = 20.17 years) watched a mock, videotaped crime. Then, following a brief delay, they viewed a 6-person target-present or -absent lineup using one of the four lineup procedures. For target-present lineups, correct identification rates for the four lineup procedures were comparable. In contrast, for target-absent lineups, the correct rejection rate was higher using the elimination lineup procedure compared to the wildcard and simultaneous lineup procedures. Remaining comparisons between lineup procedures found no significant differences. Also diagnosticity ratios were similar across the four procedures. [ABSTRACT FROM AUTHOR]

The ADC method of proof search in propositional natural deduction proposed in 2000 proceeds bottom up first by Analysing the sequent into sub-goals by applying all possible introduction rules, and then by checking whether each of these sub-goals can be established using only elimination rules (Direct Chaining). This looks simpler than the worst case complexity (PSPACE) of the derivability problem for intuitionistic propositional logic.We investigate the complexity ofADC for various fragments. ADC derivability is polynomially decidable for the &,→-fragment by a generalization of a familiar direct chaining algorithm for Horn formulas. Adding ⋁ leads to a CoNP-complete fragment. A short counterexample in case the goal sequent is not ADC derivable is provided by witnessing all relevant disjunctions. Adding constant ⊥ preserves CoNP completeness. [ABSTRACT FROM AUTHOR]

Classifying large and real-world datasets is a challenging problem in machine learning algorithms. Among the machine learning methods, the support vector machine (SVM) is a well-known approach with high generalization ability. Unfortunately, while the number of training data increases and the data contain noise, the performance of SVM signifcantly decreases. In this paper, a fast and de-noise two-stage method for training SVMs to deal with large, real- world datasets is proposed. In the frst stage, data that contain noises or are suspected to be noisy are identifed and eliminated from the genuine training dataset. The process of elimination and identifcation is based on the movement of the center of the convex hull data in the training dataset. The convex hull data are computed via the QHull algorithm. On the other hand, the well-known fuzzy clustering method (FCM) is applied to compress and reduce the size of the training dataset. Finally, the reduced and purifed cluster centers are used for training the SVM. A set of experiments is conducted on the four benchmarking datasets of the UCI database. Moreover, the amount of training time and the generalization of the proposed approach are compared with FCM-SVM and normal SVM. The results indicate that the proposed method reduces the amount of training time and has a considerable success in removing noisy data from the training dataset. Therefore, the proposed method can achieve a higher generalization performance in comparison with the other methods in large, real-world datasets. [ABSTRACT FROM AUTHOR]

Nuisance parameter elimination is a central problem in capture–recapture modelling. In this paper, we consider a closed population capture–recapture model which assumes the capture probabilities varies only with the sampling occasions. In this model, the capture probabilities are regarded as nuisance parameters and the unknown number of individuals is the parameter of interest. In order to eliminate the nuisance parameters, the likelihood function is integrated with respect to a weight function (uniform and Jeffrey's) of the nuisance parameters resulting in an integrated likelihood function depending only on the population size. For these integrated likelihood functions, analytical expressions for the maximum likelihood estimates are obtained and it is proved that they are always finite and unique. Variance estimates of the proposed estimators are obtained via a parametric bootstrap resampling procedure. The proposed methods are illustrated on a real data set and their frequentist properties are assessed by means of a simulation study. [ABSTRACT FROM AUTHOR]

Computers play a serious role in human life, especially web-based applications running twenty four hours per day. These applications are based on relational database management system and they receive many queries from the users. These queries are executed in the commercial systems one by one without any consideration of past experiences and data analysis. The execution of queries can be faster if some rules were derived from the past queries. In this paper, we propose a statistical query-based rule derivation system by the backward elimination algorithm, which analysis the data based on the past queries in order to derive new rules, and then it uses these rules for the execution of new queries. The computational results are presented and analysed that the system is very efficient and promising. [ABSTRACT FROM AUTHOR]

We define a reasonably well-behaved class of ultraimaginaries, i.e. classes modulo -invariant equivalence relations, called tame, and establish some basic simplicity-theoretic facts. We also show feeble elimination of supersimple ultraimaginaries: If is an ultraimaginary definable over a tuple with , then is eliminable up to rank . Finally, we prove some uniform versions of the weak canonical base property. [ABSTRACT FROM AUTHOR]

We generalize and unify various tools from the study of valued fields equipped with an automorphism to obtain a relative quantifier elimination result for such fields. Along the way we point how the techniques we employ relate to a classical result from tropical geometry. The quantifier elimination result we provide is then applied to the transseries field equipped with the automorphism which sends $$f(x)$$ to $$f(x+1)$$ . [ABSTRACT FROM AUTHOR]

Summary: In this paper, an investigation into the behavior of modern narrowband power‐line communication systems when concatenated with both a fountain code named Luby transform code and a combination of nonlinear preprocessing techniques called signal clipping with blanking is presented. The new systems can be used in applications such as the automatic meter reading component of smart grid technology. Testing is performed on a range of systems provided by the G3 standard and the results show that a bit error rate improvement can be achieved with the correct encoder/decoder settings. The recommended settings include an adaptive technique that uses Gaussian elimination for Luby transform code decoding and code rates that depend on the system specifications used. From there, it is shown that a further improvement is achieved when both the Luby transform code and adaptive signal clipping/blanking based on signal power are combined. It is found that these modifications present a complexity trade‐off, which is higher on the receiver side. This added complexity can be well‐tolerated in automatic meter‐reading systems because of the star topology and asymmetric nature of the network, which has a centralized agent acting as a receiver that collects data from various transmitter nodes. [ABSTRACT FROM AUTHOR]

Fingerprint positioning can take advantage of existing WLAN to achieve indoor locations, which has been widely studied. We analyzed the corresponding positions distribution of similar fingerprints, and then found that the fuzzy similarity between fingerprints is the root cause of the larger errors existing. According to clusters distribution feature of corresponding positions of the similar fingerprints, we proposed a K-Means+ clustering algorithm to achieve fine-grained fingerprint positioning. Due to the K-Means+ algorithm failing to locate the positions of outliers, we also designed a linear sequence matching algorithm to improve the outliers positioning, and reduce the impact of fuzzy similarity. Experimental results illustrate that our algorithm can get a maximum positioning error less than 5 m, which outperforms other algorithms. Meanwhile, all the positioning errors over 4 m in our algorithm are less than 2%. The positioning accuracy has been improved significantly. [ABSTRACT FROM AUTHOR]

For efficiency in a linear algebra course the instructor may wish to avoid the undue arithmetical distractions of rational arithmetic. In this paper we explore how to write fraction-free problems of various types including elimination, matrix inverses, orthogonality, and the (non-normalizing) Gram–Schmidt process. [ABSTRACT FROM AUTHOR]

Given a rational projective parametrization P(s, t, v) of a rational projective surface S we present an algorithm such that, with the exception of a finite set (maybe empty) B of projective base points of P, decomposes the projective parameter plane as P2(K) \ B = Ulk=1 бk such that, if (s0 : t0 : v0) ∊ бk, then P(s0, t0, v0) is a point of S of multiplicity k. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of finding time-optimal trajectories in two-dimensional space, where the mover's speed monotonically decreases or increases in one of the space's coordinates. We address such problems in different settings for the velocity function and in the presence of obstacles. First, we consider the problem without any obstacles. We show that the problem with linear speed decrease is reducible to the de l'Hôpital's problem, and that, for this case, the time-optimal trajectory is a circular segment. Next, we show that the problem with linear speed decreases and rectilinear obstacles can be solved in polynomial time by a dynamic program. Finally, we consider the case without obstacles, where the medium is non-uniform and the mover's velocity is a piecewise linear function. We reduce this problem to that of solving a system of polynomial equations of fixed degree, for which algebraic elimination theory allows us to solve the problem to optimality. [ABSTRACT FROM AUTHOR]

Elimination by aspects (EBA) is a random utility model that is considered to represent the choice process used by consumers more faithfully than logit and probit models. One limitation of the model is that it does not have a known error theory. We show that EBA can be derived by assuming that aspects have random utilities with independent, extreme value distributions. Multinomial logit and rank-ordered logit models are special cases of EBA. [ABSTRACT FROM AUTHOR]

We present a Poisson formula for sparse resultants and a formula for the product of the roots of a family of Laurent polynomials, which are valid for arbitrary families of supports. To obtain these formulae, we show that the sparse resultant associated to a family of supports can be identified with the resultant of a suitable multiprojective toric cycle in the sense of R'emond. This connection allows to study sparse resultants using multiprojective elimination theory and intersection theory of toric varieties. [ABSTRACT FROM AUTHOR]

The strand symmetric model is a phylogenetic model designed to re ect the symmetry inherent in the double-stranded structure of DNA. We show that the set of known phylogenetic invariants for the general strand symmetric model of the three leaf claw tree entirely defines the ideal. This knowledge allows one to determine the vanishing ideal of the general strand symmetric model of any trivalent tree. Our proof of the main result is computational. We use the fact that the Zariski closure of the strand symmetric model is the secant variety of a toric variety to compute the dimension of the variety. We then show that the known equations generate a prime ideal of the correct dimension using elimination theory. [ABSTRACT FROM AUTHOR]

Connexive logic contains propositional theses such as NCpNp and CCpqNCpNq that are false in two-valued logic. It derives from remarks made by Sextus Empiricus in the fourth century B.C. The logic of causal and subjunctive conditionals is also connexive, since ‘If X is dropped, it will hit the floor’ contradicts ‘If X is dropped, it will not hit the floor’. In the paper below, a Gentzen Sequenzenkalkul formulation of connexive logic is constructed, and a proof is given of the elimination theorem. [ABSTRACT FROM AUTHOR]

The rigidity of a matrix A for target rank r is the minimum number of entries of A that must be changed to ensure that the rank of the altered matrix is at most r. Since its introduction by Valiant (), rigidity and similar rank-robustness functions of matrices have found numerous applications in circuit complexity, communication complexity, and learning complexity. Almost all n × n matrices over an infinite field have a rigidity of ( n − r). It is a long-standing open question to construct infinite families of explicit matrices even with superlinear rigidity when r = Ω( n). In this paper, we construct an infinite family of complex matrices with the largest possible, i.e., ( n − r), rigidity. The entries of an n × n matrix in this family are distinct primitive roots of unity of orders roughly exp( n log n). To the best of our knowledge, this is the first family of concrete (but not entirely explicit) matrices having maximal rigidity and a succinct algebraic description. Our construction is based on elimination theory of polynomial ideals. In particular, we use results on the existence of polynomials in elimination ideals with effective degree upper bounds (effective Nullstellensatz). Using elementary algebraic geometry, we prove that the dimension of the affine variety of matrices of rigidity at most k is exactly n - ( n - r) + k. Finally, we use elimination theory to examine whether the rigidity function is semicontinuous. [ABSTRACT FROM AUTHOR]

Nonnegative tensor factorization has applications in statistics, computer vision, exploratory multiway data analysis, and blind source separation. A symmetric nonnegative tensor, which has an exact symmetric nonnegative factorization, is called a completely positive tensor. This concept extends the concept of completely positive matrices. A classical result in the theory of completely positive matrices is that a symmetric, diagonally dominated nonnegative matrix is a completely positive matrix. In this paper, we introduce strongly symmetric tensors and show that a symmetric tensor has a symmetric binary decomposition if and only if it is strongly symmetric. Then we show that a strongly symmetric, hierarchically dominated nonnegative tensor is a completely positive tensor, and present a hierarchical elimination algorithm for checking this. Numerical examples are given to illustrate this. Some other properties of completely positive tensors are discussed. In particular, we show that the completely positive tensor cone and the co-positive tensor cone of the same order are dual to each other. [ABSTRACT FROM AUTHOR]