Qa := ∫ Qa ⊗ P (xj|b,b ′,x, a,y) dxj [Symbolic Substitution] For all bi in Qa Qa := [ Qa ⊗ P (bi|b,x, a,y) ] |b′i=1 ⊕ [ Qa ⊗ P (bi|b,x, a,y) ] |b′i=0 [Case ⊕] Compute final Q-Value (discount and add reward): Qa := R(b,x, a,y)⊕ (γ ⊗Qa) Note that ∫ f (xj)⊗δ[xj−h(z)]dxj = f (xj){xj/h(z)}where the latter operation indicates that any occurrence of xj in f (x ′ j) is symbolically substituted with the...

We present two multiscale reaction-diffusion (RD) systems modeling sulfate attack in concrete structures (here: sewer pipes). The systems are posed on two different spatially separated scales. The only difference between them is the choice of the micro-macro transmission condition. We explore numerically the way the macroscopic Biot number Bi connects the two reactiondiffusion scenarios. We ind...

Let H be a Hopf algebra with bijective antipode over a field k and suppose that R#H is a bi-product. Then R is a bialgebra in the Yetter–Drinfel’d category HYD. We describe the bialgebras (R#H) and (R#H) explicitly as bi-products R#Hop and R#H respectively where R is a bialgebra in H op HopYD and R o is a bialgebra in H o HoYD. We use our results to describe two-cocycle twist bialgebra structur...

Bis(amidodimethyl)disiloxane ligands derived from O{SiMe2N(H)R}2 (abbreviated as (NONR)H2) are a stable support for neutral and cationic bismuth compounds. Attempts to extend the series Bi(NONR)Cl (R = Ar = 2,6-iPr2C6H3; R = tBu) to include compounds where R = Ar' = 2,6-Me2C6H3 were complicated by concomitant formation of the bimetallic compound {Bi(NONAr')}2(μ-NONAr'). Compounds containing [Bi...

MOTIVATIONS Bi-clustering is an important approach in microarray data analysis. The underlying bases for using bi-clustering in the analysis of gene expression data are (1) similar genes may exhibit similar behaviors only under a subset of conditions, not all conditions, (2) genes may participate in more than one function, resulting in one regulation pattern in one context and a different patte...

Topological insulators and graphene present two unique classes of materials, which are characterized by spin-polarized (helical) and nonpolarized Dirac cone band structures, respectively. The importance of many-body interactions that renormalize the linear bands near Dirac point in graphene has been well recognized and attracted much recent attention. However, renormalization of the helical Dir...

The explosion of biological data has largely influenced the focus of today's biology research. Integrating and analysing large quantity of data to provide meaningful insights has become the main challenge to biologists and bioinformaticians. One major problem is the combined data analysis of data from different types, such as phenotypes and genotypes. This data is modelled as bi-partite graphs ...

This paper presents new efficient methods to find “optimal bi-decomposition” forms of logic functions. An “optimal bi-decomposition” form of f(X) is f = α(g1(X ), g2(X )) where the total number of variables in X and X is the smallest among all bi-decomposition forms of f . We consider two methods; one’s decomposition form is (g1 · g2) and the other’s is (g1⊕g2). The proposed methods can find on...

For many design applications, where multiple primary surface pieces meet, the distribution of curvature is more important than formally achieving exact curvature continuity. For parametric spline surfaces, when constructing a multi-sided surface cap, we demonstrate a strong link between the uniform variation of the re-parameterization between (boundary) data of the joining pieces and a desirabl...

This paper studies the relationship between the so-called bi-quadratic optimization problem and its semidefinite programming (SDP) relaxation. It is shown that each r -bound approximation solution of the relaxed bi-linear SDP can be used to generate in randomized polynomial time an O(r)-approximation solution of the original bi-quadratic optimization problem, where the constant in O(r) does not...