Elements Xi Research Articles

Linear Maps Which are Anti-derivable at Zero

Let T:Arightarrow X be a bounded linear operator, where A is a hbox {C}^*-algebra, and X denotes an essential Banach A-bimodule. We prove that the following statements are equivalent: (a)T is anti-derivable at zero (i.e., ab =0 in A implies T(b) a + b T(a)=0);(b)There exist an anti-derivation d:Arightarrow X^{**} and an element xi in X^{**} satisfying xi a = a xi ,xi [a,b]=0,T(a b) = b T(a) + T(b) a - b xi a, and T(a) = d(a) + xi a, for all a,bin A. We also prove a similar equivalence when X is replaced with A^{**}. This provides a complete characterization of those bounded linear maps from A into X or into A^{**} which are anti-derivable at zero. We also present a complete characterization of those continuous linear operators which are ^*-anti-derivable at zero.

Studying the inertias of LCM matrices and revisiting the Bourque-Ligh conjecture

Let S={x1,x2,…,xn} be a finite set of distinct positive integers. Throughout this article we assume that the set S is GCD closed. The LCM matrix [S] of the set S is defined to be the n×n matrix with lcm(xi,xj) as its ij element. The famous Bourque-Ligh conjecture used to state that the LCM matrix of a GCD closed set S is always invertible, but currently it is a well-known fact that any nontrivial LCM matrix is indefinite and under the right circumstances it can be even singular (even if the set S is assumed to be GCD closed). However, not much more is known about the inertia of LCM matrices in general. The ultimate goal of this article is to improve this situation. Assuming that S is a meet closed set we define an entirely new lattice-theoretic concept by saying that an element xi∈S generates a double-chain set in S if the set meetcl(CS(xi))∖CS(xi) can be expressed as a union of two disjoint chains (here the set CS(xi) consists of all the elements of the set S that are covered by xi and meetcl(CS(xi)) is the smallest meet closed subset of S that contains the set CS(xi)). We then proceed by studying the values of the Möbius function on sets in which every element generates a double-chain set and use the properties of the Möbius function to explain why the Bourque-Ligh conjecture holds in so many cases and fails in certain very specific instances. After that we turn our attention to the inertia and see that in some cases it is possible to determine the inertia of an LCM matrix simply by looking at the lattice-theoretic structure of (S,|) alone. Finally, we are going to show how to construct LCM matrices in which the majority of the eigenvalues is either negative or positive.

Random Walk in Balls and an Extension of the Banach Integral in Abstract Spaces

We describe the construction of a random walk in a Banach space mathbb {B} with a quasi-orthogonal Schauder basis and show that it is a martingale. Next we prove that under certain additional assumptions the described random walk converges a.s. and in L^{p}left( mathbb {B}right) ,,1le p,<infty , to a random element xi , which generates a probability measure with support contained in the unit ball Bsubset mathbb {B}. Moreover, we define the Banach integral with respect to the distribution of xi for a class of bounded, Borel measurable real-valued functions on B. Next some examples of nonstandard Banach spaces with quasi-orthogonal Schauder bases are presented; furthermore, examples which demonstrate the possibility of applications of all the obtained results in spaces ell ^{p},,1le p < infty and L^{p}[0,1],,1< p < infty are given.

The Walking Dead Live!: Essays on the Television Show Philip L.Simpson and MarcusMallard, Editors. Rowman &amp; Littlefield, 2016.

The Walking Dead Live!: Essays on Television Show Philip L. Simpson and Marcus Mallard, Editors. Rowman & Littlefield, 2016.One could hardly question rise in popularity of zombies in American culture in past decade: TV, literature, how-to guides, video games, graphic novels, feature films, zombie runs, and merchandise. The AMC network's hypersuccessful series The Walking Dead leads charge, and it has even produced a spin-off prequel, Fear Walking Dead. Analyses of zombies and accompanying apocalypse have evolved into a productive, interdisciplinary critique of American and global economics, psychology, sociology, philosophy, ecology, gender, and race. Although zombie has been and still is under analysis for its symbolic elements, critique is shifting toward us-for example, how are we like zombies? why are we so entranced by zombie narratives at this point in time? The last two seasons of The Walking Dead have attracted over seventeen and fourteen million viewers respectively (ix), and as series enters its Season Seven, volume and delivery of material to analyze are unique compared to individual films and literary texts. The Walking Dead Live!: Essays on Television Show, edited by Philip L. Simpson and Marcus Mallard, illustrates depth and breadth in which critical analysis has also evolved to parallel zombie's stature in American culture.The editors cite five other anthologies specific to The Walking Dead (series and graphic novels) and separate their anthology by [addressing] issues that have either not been raised yet ... or contribute to and expand upon ongoing conversations about specific story and subtextual elements (xi). The anthology is evenly divided into Part One, which centers on series's central conceits, philosophies, genre influences, and story elements (xii), and Part Two, which hones in on the survivors' quest to find a haven in which to rebuild human society (xii). Overall, collection serves as a useful spectrum of critical techniques applied to a television series that continues to evolve.The first essay, 'Zombies R Us': Twenty-First Century America and Historical Inquiry by Brad L. Duren, provides a general historical view of zombie in American culture and highlights importance of analyzing pop culture, and essay is well placed to offer readers new to zombie critique an entry point. A few of contributions are very specific to a particular theorist or approach, and these would be helpful to scholars or students looking for new critical applications. …

On iterative methods for solving equations with covering mappings

In this paper we propose an iterative method for solving the equation Υ(x, x) = y, where the mapping Υ acts in metric spaces and is covering in the first argument and Lipschitzian in the second one. Each subsequent element xi+1 of the sequence of iterations is defined by the previous one as a solution to the equation Υ(x, xi) = yi, where yi can be an arbitrary point sufficiently close to y. Conditions for convergence and error estimates are obtained. The method proposed is an iterative development of the Arutyunov method for finding coincidence points of mappings. In order to determine xi+1 in practical implementation of the method in linear normed spaces, it is proposed to perform one step by using the Newton–Kantorovich method. The thus-obtained method of solving the equation of the form Υ(x, u) = ψ(x) − φ(u) coincides with the iterative method proposed by A.I. Zinchenko,M.A. Krasnosel’skii, and I.A. Kusakin.

Complex Field Fault Modeling-Based Optimal Frequency Selection in Linear Analog Circuit Fault Diagnosis

Owing to the lack of feasible fault modeling method, hard (open and short) faults, and discretized parametric faults are still the mostly used fault models. These models cannot characterize all soft (parameter shifting) faults because that the parameter of analog element be of continuity character. To address this concern, a complex field fault modeling method is presented first. If fault happens to passive element xi in linear analog circuit, the real (Ur) and imaginary (Uj) parts of faulty voltage phasor U°(U°=Ur+j Uj) must satisfy binary quadratic function Fi(Ur,Uj)=0, which is independent from the value of element xi and uniquely determined by its location and the nominal circuit under test. Hence, the binary quadratic function can model any continuous parameter shifting (soft) or hard fault. Second, to avoid calculating the explicit expression of binary quadratic function, a simulation-based method is given to obtain the locus of the change in the function Fz in the complex plane. Besides, the nominal (fault-free) point, the loci determined by the binary quadratic functions might intersect with each other. It is referred to as aliasing problem in this paper. This problem can be solved by adjusting the frequency of the input signal. Therefore, a constraint optimization method is proposed to select optimal test frequencies. The proposed test generation and compression methods, when combined with the complex field modeling method, can pinpoint much more fault components with less test points and frequencies when compared with the other methods.

Predicting the Production of Total Industry in Greece with Chaos Theory and Neural Networks

1. Introduction Non linear dynamics (Medio, 1992) in combination with neural networks had applied in a wide variety of fields, e.g. physics, engineering, ecology and economics. The economist interest is focused on the ability of forecasting an economic time series using time series analysis (Thalassinos and Pociovalisteanu, 2007). In this work we have applied non linear time series analysis in monthly values of Greece Total Industry index. We cover time period from 01.01.1962 until 01.04.2011. We have applied the method of Grassberger and Procaccia (Grassberge and Procaccia, 1983a and 1983b) to evaluate the minimum embedding dimension of each the system. In a second stage using the neural network (Hanias, Curtis, Thalassinos, 2007; Thalassinos et al., 2008 and 2009) we achieved an out of sample multi step time series prediction. 2. Time Series The data for the Production of Total Industry in Greece are collected from Organization for Economic Co-operation and Development and presented as a signal x=x(t) as it shown at Figure 1 (01.01.1962 - 01.04.2011). The sampling rate is At=1 month and the number of data are N=592. 3 [FIGURE 1 OMITTED] 3. State Space Reconstruction For a scalar time series, in our case the time series is the Production of Total Industry index, the phase space can be reconstructed using the methods of delays. The basic idea in the method of delays is that the evolution of any single variable of a system is determined by the other variables, with which it interacts. Information about the relevant variables is thus implicitly contained in the history of any single variable. On the basis of this an equivalent phase space can be reconstructed. From our data we construct a vector [[bar.X].sub.i], i=1 to N, in the m dimensional phase space given by the following relation (Kantz and Schreiber, 1997; Takens, 1981). [[bar.X].sub.i] = {xi,xi-[tau],xi-2[tau], .... xi+(m-1)x}[tau] (1) This vector represents a point to the m dimensional phase space in which the attractor is embedded each time, where [tau] is the time delay [tau]=i[DELTA]t. The element xi represents a value of the examined scalar time series in time corresponding to the ith component of the time series. Use of this method, reduces phase space reconstruction to the problem of proper determining suitable values of m and t. The choice of these values is not always simple, especially when we do not have any additional information about the original system and the only source of data is a simple sequence of scalar values, acquired from the original system. The dimension, where a time delay reconstruction of the phase space provides a necessary number of coordinates to unfold the dynamics from overlaps on itself caused by projection, is called embedding dimension m. a. Time delay [tau] Using the average mutual information we can obtain x less associated with linear point of view, and thus more suitable for dealing with nonlinear problems. The average mutual information may be expressed by the following formula (Kantz and Schreiber, 1997; Takens, 1981). [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2) where P(xi) represents probability of value xi and P(xi, xi+[tau]) is joint probability. In general, I([tau]) expresses the amount of information (in bits), which may be extracted from the value in time xi about the value in time xi+[tau]. As [tau], suitable for the phase space reconstruction, is the first minimum of I(t). [FIGURE 2 OMITTED] As shown in Figure 2, in our case the mutual information function I(t) exhibits a local minimum at 4 time steps and, thus, we shall consider [tau] = 4 to be the optimum delay time. b. Embedding dimension m One method to determine the presence of chaos is to calculate the fractal dimension, which will be non integer for chaotic systems. Even though there exists a number of definitions for the dimension of a fractal object (Box counting dimension, Information Dimension, etc. …

Cryptosystems based on semi-distributive algebras

We propose a new cryptographic scheme of ElGamal type. The scheme is based on algebraic systems defined in the paper—semialgebras (Sect. 2). The main examples are semialgebras of polynomial mappings over a finite field K, and their factor-semialgebras. Given such a semialgebra R, one chooses an invertible element a ∈ R* of finite order r, and a random integer s. One chooses also a finite dimensional K-submodule V of R. The 4-tuple (R, V, a, b) where b = as forms the public key for the cryptosystem, while r and s form the secret key. A plain text can be viewed as a sequence of elements of the field K. That sequence is divided into blocks of length dim(V) which, in turn, correspond to uniquely determined elements Xi of V. We propose three different methods (A, B, and C, see Definition 1.1) of encoding/decoding the sequence of Xi. The complexity of cracking the proposed cryptosystem is based on the Discrete Logarithm Problem for polynomial mappings (see Sect. 1.1). No methods of cracking the problem, except for the “brute force” (see Sect. 1.1) with Ω(r) time, are known so far.

Calculation of liquidus temperature of low and high alloyed iron base melts from their chemical composition by means of the equivalence factors

Liquidus temperature of a steel is calculated from its chemical composition according to the so far proposed procedures by simply summing up the iron melting point lowering effects of all in the steel contained elements Xi, as they are given by approximating the iron liquidus curves with the straight lines in the corresponding binary Fe‐Xi systems. The intensification of the iron melting point lowering effect with the increasing concentration of each element contained in the steel has not been at all or only partly taken into account in this approach. This is the reason why the calculated steel liquidus temperatures are not accurate enough.In the here proposed new calculation procedure the iron melting point lowering effect of an alloying element Xi is first translated into an equivalent effect of carbon in the Fe‐C system by converting the concentration of Xi into an equivalent concentration of carbon taken as a reference element. The carbon equivalent iron melting point lowering effect of all elements contained in the steel including carbon is then evaluated by summing up the calculated equivalent carbon concentrations of all elements together with carbon into a fictitious total equivalent carbon concentration. A so calculated total equivalent carbon concentration has made it possible to determine with a good accuracy the steel liquidus temperature along the stable δ/α or γ liquidus lines in the Fe‐C system. The success of a melting point calculation is strongly dependent on the accuracy of the equivalence factor equations for each alloying element. They are derived by relating the iron liquidus line in the binary Fe‐Xi system to that in the Fe‐C phase diagram.

On Infinite Locally Finite Groups

AbstractIf G is a group such that every infinite subset of G contains a commuting pair of elements then G is centre-by-finite. This result is due to B. H. Neumann. From this it can be shown that if G is infinite and such that for every pair X, Y of infinite subsets of G there is some x in X and some y in Y that commute, then G is abelian. It is natural to ask if results of this type would hold with other properties replacing commutativity. It may well be that group axioms are restrictive enough to provide meaningful affirmative results for most of the properties. We prove the following result which is of similar nature.If G is a group such that for each positive integer n and for every n infinite subset X1,...,Xn of G there exist elements xi of Xii = 1,... ,n, such that the subgroup generated by {x1,... ,xn} is finite, then G is locally finite.

Locally inner automorphisms of CC-groups

Groups with Cernikov conjugacy classes, or CC-groups, were first considered by Polovickii [9, lo] as an extension of the concept of FC-groups. A group G is said to be a CC-group if G/C&xc) is a Cernikov group for each x E G. Polovickii’s basic result is that G is a CC-group if and only if the normal closure (xc) of each element of G is Cernikov-by-cyclic and G/C,(x’) is periodic for each x E G. It follows that the periodic CC-groups are the groups which are locally (normal and Cernikov) in the sense that they have a local system consisting of normal Cernikov subgroups. An automorphism cp of a group G is said to be locally inner if, for each finite set of elements xi, . . . . x, E G, there is an element gE G such that xi’P=g -lxi g, for i = 1, . . . . n. The locally inner automorphisms of G clearly form a subgroup of Aut G, which we denote by Linn G. Two subgroups H

Generalizations of Reductions and Mixed Multiplicities

The first section of this paper is devoted to the definition, and proof of the existence, of what are called complete and joint reductions of a set of ideals of a d-dimensional local ring Q with maximal ideal m and residue field k. The former is defined for a set of ideals a1,…, as of Q, and consists of a set of sd elements xij (i = l,…,s;j = 1, …,d) where xij ⊂ ai and the elements yj = xljx2j… xsj(j = 1,…, d) form a reduction of a1 a2… as. The definition of a joint reduction applies to a set of, not necessarily distinct, ideals a1,…, ad, and d s a set of elements xi (i = 1,…, d) such that ∑ d i = 1 xia1…ai=1ai+1…ad is a reduction of a1a2…ad. The second section commences with a treatment of Hilbert functions of several m-primary ideals based on deas of Buchsbaum and Auslander [1, corrections], and sketched in [8]. This is used to prove that the multiplicity of a joint reduction of d m-primary ideals a1,…,ad depends only on a1,…,ad and is a mixed multiplicity as defined by Teissier in [9]. The final part of Section 2 is devoted to a proof of the following result. Suppose that Q is analytically unramified, and that L(Q / (an)′), where (an)′ is the integral closure of an, with a m-primary, is equal to the polynomial (e(a)nd/d!−½(f(a)nd−1/(d−1)!)+… for large n. Then f(a) is a homogeneous polynomial over the set of m-primary ideals of Q in the sense that f ( a 1 r 1 ⋯ a s r s ) can be expressed as a homogeneous polynomial of degree d − 1 for r1,…, rs ⩾ 0(and not merely for r1,…, rs all positive). This extends a result proved in the case d = 2 in [7] to general d.

Relative normal complements in finite groups

( G , H , H 0 , π ) (G,H,{H_0},\pi ) denotes the following configuration: H H and H 0 {H_0} are the subgroups of the finite group G G with H 0 ⊴ H {H_0} \trianglelefteq H is the set of primes dividing ( H : H 0 ) (H:{H_0}) . For ( G , H , H 0 , π ) (G,H,{H_0},\pi ) we consider conditions ( A ) ({\text {A}}) , ( B 0 ) ({{\text {B}}_0}) , and ( C ) ({\text {C}}) : ( A ) ({\text {A}}) Any two π \pi -elements of H − H 0 H - {H_0} which are G G -conjugate are H H -conjugate. ( B 0 ) ({{\text {B}}_0}) For each π \pi -element x ∈ H − H 0 x \in H - {H_0} , C G ( x ) = I ( x ) C H ( x ) {C_G}(x) = I(x){C_H}(x) where I ( x ) I(x) is a normal π ′ \pi ’ -subgroup of C G ( x ) {C_G}(x) . ( C ) | ( H − H 0 ) G , π | = ( G : H ) | H − H 0 | ({\text {C}})\left | {{{(H - {H_0})}^{G,\pi }}} \right | = (G:H)\left | {H - {H_0}} \right | . We show that if ( G , H , H 0 , π ) (G,H,{H_0},\pi ) satisfies ( B 0 ) ({{\text {B}}_0}) and ( C ) ({\text {C}}) , or ( A ) ({\text {A}}) and ( B 0 ) ({{\text {B}}_0}) , and if H / H 0 H/{H_0} is solvable, then there is a unique relative normal complement G 0 {G_0} of H H over H 0 {H_0} .

On the symmetric and Rees algebra of an ideal generated by a d-sequence

Let R be a commutative ring and I an ideal of R. In this paper, we consider the question of when the symmetric algebra of I is a domain, and hence isomorphic to the Rees algebra of I. (see Section 2 for definitions.) Several authors have studied this question (for example, [I, 4, 9, lo], or [14]). In the cases in which the symmetric algebra is a domain, other questions have been asked: Is it Cohen-Macaulay [l] ? Is it factorial [15] ? Is it integrally closed [1, 121 ? In this paper we prove the symmetric algebra of I is a domain whenever R is a domain and I is generated by a d-sequence (see [6] or [7]). A sequence of elements xi ,..., x, in R is said to be a d-sequence if (i) xi 4 (x1 ,..., xiwl , xi+r ,..., x,) for i between 1 and n and (ii) if {il ,..., ii} is a subset (possibly (6) of u,..., TZ} and K, rn~ { l,..., n}\(il ,..., ii} then((xil ,..., xi,) : xlcx,) = ((xi, ,..., xi,) : xlc). Many examples were given in [7] of d-sequences. We list some examples here.

ON THE WITT GROUP OF A 2-ADIC GROUP RING

Z2TT, where -n is a 2-group, endowed with the orientable involution f = g~1. In this paper we explicitly calculate the Witt group W(A) of Hermitian forms over A with respect to this involution. The difficult part of the calculation is the quotient W(A)/W™(A), where W°(A) denotes the subgroup of even Hermitian forms. To obtain W(A), one must observe that Wv(A)ssZ/2 and is a direct summand of W(A). This was pointed out by W. Pardon. The calculation is a generalization of the method of characteristi c elements, which one uses to compute W(Z)/W™(Z)ssZ/8. (See [4].) There one associates to every non-singular bilinear form over Z a characteristic element x in the underlying free Z-module and observes that {x, x) is an invariant of the isomorphism class of the form if reduced mod 8. Moreover, this invariant vanishes on even forms, hence the above isomorphism. In the case of A = Z2ir, we find that we must associate a collection of characteristi c elements Xu • • • > Xk> m tne underlying free A-module, one for each conjugacy class of involutions in n. Furthermore, the elements (xi, xj) are aU invariants in suitable quotient groups of A, and most of the work in this paper is the evaluation of the quotient groups. These invariants detect all but a small subgroup of W(A)/W°(A), i.e., the subgroup of Hermitian forms for which all these invariants vanish is isomorphic to TT^, ® ZJ2, where TI^, is the commutator quotient TT/[TT, IT].

On products of powers in groups

In this note we show that a product of N N h powers in a group cannot in general be expressed as a product of fewer N N th powers. This extends a result of Lyndon and Newman [1].

Corrections to Codimension and Multiplicity

1. In proving Proposition 3.5 on page 637, we assumed tacitly that R has finite length. The statement of the proposition is correct as given, i.e., without the additional hypothesis that R has finite length. A proof for this more general situation can be obtained merely by applying induction on s and Theorem 3.3 immediately following the first paragraph of the proof of 3.5. 2. The second paragraph of Theorem 5.6 on page 646 should read as follows: Let E be an R-module having the following properties: (a) dimE= dimR. (b) For every set of elements xi, *, xk in m such that rank (E/(x1, *, xE) = k, the submodule (0) in E/(x1, * * *, xk)E is unmixed. If a set of elements z1, ., Zs in m (s = dim R) can be found such that (E/(z1, I, z)E) = k for 1 n and so codimE > s n .

THE STRUCTURE OF PRIME RINGS WITH MAXIMUM CONDITIONS

course that (i,,(c))* is an integral domain. Finally, since (eS (c))* is a finite )dule over the complete local domain(,,()* and has no zero-divisors, it must a complete local domain.4 This completes the proof of our theorem. We shall conclude this note by a few remarks on the hypothesis of our theorem. observe that in our formulation of the local Theorem of Bertini, we have ;roduced the subring e and a system of regular parameters x,, ..., xt in , lile in the final assertion of the theorem only the three elements xi, x2, X3 enter ;o the picture. The question naturally arises whether it is necessary at all to ;roduce the regular local domain ( or even the system of parameters xl, ..., in t. As to the latter, one finds a measure of justification in the fact that, en in the equal-characteristic case where T9 contains a coefficient field k, an 9itrary set of three analytically independent elements xl, x2, X3 may generate an ideal of rank less than 3 and hence cannot be extended into a system of rameters in ?. This is a well-known situation where a local ring may contain as ;ubring a local ring of higher dimension, and one can easily obtain examples to ow that the local Theorem of Bertini cannot hold for an arbitrary set of three

An extension of the “principal theorem” of Wedderburn

By considering elements whose principal traces are zero instead of nilpotent elements, this paper exhibits an ideal ZD91 called the liberal, which is a generalization of the radical and which, in fact, reduces to the radical when and only when 2191 is separable. An extension of Theorem 1 is obtained in which there is always a decomposition of type (1) for the liberal of an arbitrary alternative algebra. The associative case of this paper is a portion of the author's doctoral dissertation written under Professor C. C. MacDuffee. 2. The first liberal. Let t= E#;ej be a general element [2, p. 112] of an alternative algebra with basis e1, e2, * * , en over an arbitrary field 0. Then T(t), the negative of the coefficient of the second highest power of X in the minimum polynomial m(t, X) of {, is called the principal trace of t. When the indeterminates {i are replaced by elements xi of 0 in T(t) we get the principal trace T(x) of x = Exiei. The quantity T(x) is independent of the basis of Wf. We now wish to prove the following lemma.

On induced permutation matrices and the symmetric group

The n! operations Ai of permutations upon n different ordered symbols correspond to n! matrices Ai of the nth order, which have in each row and in each column only one non-zero element, namely a unit. Such matrices Ai are called permutation matrices, since their effect in premultiplying an arbitrary column vector x = {x1x2….xn} is to impress the permutation Ai upon the elements xi. For example the six matrices of the third orderare permutation matrices. It is convenient to denote them bywhere the bracketed indices refer to the permutations of natural order. Clearly the relation Ai Aj = Ak entails the matrix relation AiAj = Ak; in other words, the n! matrices Ai, give a matrix representation of the symmetric group of order n!.