We consider two-class linear classification in a high-dimensional, small-sample-size setting. Only a small fraction of the features are useful, these being unknown to us, and each useful feature contributes weakly to the classification decision. This was called the rare/weak (RW) model in our previous study (Donoho, D. & Jin, J. 2008 Proc. Natl Acad. Sci. USA 105, 14 790-14 795). We select feat...

Let R be a ring with identity 1. We say that R satisfies unit 1-stable range in case ax+b = 1 with a,x,b ∈ R implying that a+bu ∈ U(R). Many authors studied unit 1-stable range such as those of [1, 2, 3, 4, 5, 6]. Following the authors, a ring R satisfies unit 1-stable range for an ideal I provided that ax+b = 1 with a∈ I, x,b ∈ R implying that x+ub ∈U(R) for some unit u∈U(R). Let R = Z/2Z⊕Z/3Z...

‎let $r$ be a $*$-prime ring with center‎ ‎$z(r)$‎, ‎$d$ a non-zero $(sigma,tau)$-derivation of $r$ with associated‎ ‎automorphisms $sigma$ and $tau$ of $r$‎, ‎such that $sigma$‎, ‎$tau$‎ ‎and $d$ commute with $'*'$‎. ‎suppose that $u$ is an ideal of $r$ such that $u^*=u$‎, ‎and $c_{sigma,tau}={cin‎ ‎r~|~csigma(x)=tau(x)c~mbox{for~all}~xin r}.$ in the present paper‎, ‎it is shown that...

‎Let $R$ be a $*$-prime ring with center‎ ‎$Z(R)$‎, ‎$d$ a non-zero $(sigma,tau)$-derivation of $R$ with associated‎ ‎automorphisms $sigma$ and $tau$ of $R$‎, ‎such that $sigma$‎, ‎$tau$‎ ‎and $d$ commute with $'*'$‎. ‎Suppose that $U$ is an ideal of $R$ such that $U^*=U$‎, ‎and $C_{sigma,tau}={cin‎ ‎R~|~csigma(x)=tau(x)c~mbox{for~all}~xin R}.$ In the present paper‎, ‎it is shown that if charac...

For a field of definition k of an abelian variety A and prime ideal p of k which is of a good reduction for A, the structure of A(Fp) as abelian group is: (1) A(Fp) ' Z/d1(p)Z⊕ · · · ⊕ Z/dg(p)Z⊕ Z/e1(p)Z⊕ · · · ⊕ Z/eg(p)Z, where di(p)|di+1(p), dg(p)|e1(p), and ei(p)|ei+1(p) for 1 ≤ i < g. We are interested in finding an asymptotic formula for the number of prime ideals p with Np < x, A has a go...

Quasi-ideals were introduced by Otto Steinfeld [43] as those non-empty subsets Q of a semigroup T satisfying QTD TQ c Q. When T is regular they are precisely the subsets Q of T which satisfy QTQ = Q ([43, Theorem 9.3]). There are many examples of quasi-ideals in regular semigroup theory. We list below some of the most important: • Every subsemigroup of the form eSe (where e is an idempotent) is...

The ideal-cache model, an extension of the RAM model, evaluates the referential locality exhibited by algorithms. The ideal-cache model is characterized by two parameters-the cache size Z, and line length L. As suggested by its name, the ideal-cache model practices automatic, optimal, omniscient replacement algorithm. The performance of an algorithm on the ideal-cache model consists of two meas...

We study an interesting class of Banach function algebras of innitely dierentiable functions onperfect, compact plane sets. These algebras were introduced by Honary and Mahyar in 1999, calledLipschitz algebras of innitely dierentiable functions and denoted by Lip(X;M; ), where X is aperfect, compact plane set, M = fMng1n=0 is a sequence of positive numbers such that M0 = 1 and(m+n)!Mm+n ( m!Mm)...

Let k be an algebraically closed field. We will employ the following notation. If I ⊂ k[X1, . . . , Xn] is an ideal, we let Z(I) denote the affine algebraic set in An defined by the vanishing of the polynomials in I . Conversely, if X is an affine algebraic set, I(X) denotes the ideal of polynomials in k[X1, . . . , Xn] vanishing on X . We will give a proof of the following result, called the w...

Fins. I and z: FIG. I (left). Calculated mean quadratic elongation (h)-bond angle variance (~) distributions for four ideal types of distortion in coordination octahedra arising from the progressive shortening of the octahedral edges emphasized in insets a, b, c, and d: RGR, trend line for observed data (Robinson et al., I97I). FIG. 2 (right). Calculated mean quadratic elongation (;~)-bond angl...