Algebraic

The notion of metabolic closure is presented and analyzed in terms of Robert Rosen's theory of (M, R) systems. Recent results concerning (M, R) systems are reviewed, specially those defining self-referential equations like f (f)= f. We related (M, R) systems to ...

We prove that slices of the unitary spread of Q + (7, q), q ≡ 2 (mod 3), can be partitioned into five disjoint classes. Slices belonging to different classes are non-equivalent under the action of the subgroup of P ΓO + (8, q) fixing the unitary spread. When q is even, there is a connection between spreads of Q + (7, q) and symplectic 2-spreads of P G(5, q) (see and [8]). We determine all possible non-equivalent symplectic 2-spreads arising from the unitary spread of Q + (7, q), q = 2 2h+1 . Some of these already appeared in . When q = 3 h we classify, up to the action of the stabilizer in P ΓO(7, q) of the unitary spread of Q(6, q), those among its slices producing spreads of the elliptic quadric Q − (5, q).

Let G be a graph of order n such that $\sum_{i=0}^{n}(-1)^{i}a_{i}\lambda^{n-i}$ and $\sum_{i=0}^{n}(-1)^{i}b_{i}\lambda^{n-i}$ are the characteristic polynomials of the signless Laplacian and the Laplacian matrices of G, respectively. We show that a i ≥b i for i=0,1,…,n. As a consequence, we prove that for any α, 0<α≤1, if q 1,…,q n and μ 1,…,μ n are the signless Laplacian and the Laplacian eigenvalues of G, respectively, then $q_{1}^{\alpha}+\cdots+q_{n}^{\alpha}\geq\mu_{1}^{\alpha}+\cdots+\mu _{n}^{\alpha}$ .

The notion of metabolic closure is presented and analyzed in terms of Robert Rosen’s theory of (M,R) systems. Recent results concerning (M,R) systems are reviewed, specially those defining self-referential equations like f(f) = f. We related (M,R) systems to Autopoiesis, another theory centered on metabolic closure, and we speculate how an algebraic view of metabolism could be utilized in the analysis of metabolic pathways. As a helping metaphor we consider necessary to build, in the context of (M,R) systems, analogous to Tellegen’s theorem, but based purely on closure arguments, to metabolic networks.

We examine graphs that contain a non-trivial link in every embedding into real projective space, using a weaker notion of unlink than was used in . We call such graphs intrinsically linked in RP 3 . We fully characterize such graphs with connectivity 0,1 and 2. We also show that only one Petersen-family graph is intrinsically linked in RP 3 and prove that K 7 minus any two edges is also minor-minimal intrinsically linked. In all, 594 graphs are shown to be minor-minimal intrinsically linked in RP 3 . arXiv:0809.0454v1 [math.GT]

We point out that the positivity of a Littlewood-Richardson coefficient c γ α,β for sl n can be decided in strongly polynomial time. This means that the number of arithmetic operations is polynomial in n and independent of the bit lengths of the specifications of the partitions α, β, and γ , and each operation involves numbers whose bitlength is polynomial in n and the bit lengths α, β, and γ. Secondly, we observe that nonvanishing of a generalized Littlewood-Richardson coefficient of any type can be decided in strongly polynomial time assuming an analogue of the saturation conjecture for these types, and that for weights α, β, γ , the positivity of c 2γ 2α,2β can (unconditionally) be decided in strongly polynomial time.

We introduce a family of planar regions, called Aztec diamonds, and study tilings of these regions by dominoes. Our main result is that the Aztec diamond of order n has exactly 2n(n+1)/2 domino tilings. In this, the first half of a two-part paper, we give two proofs of this formula. The first proof exploits a connection between domino tilings and the alternating-sign matrices of Mills, Robbins, and Rumsey. In particular, a domino tiling of an Aztec diamond corresponds to a compatible pair of alternating-sign matrices. The second proof of our formula uses monotone triangles, which constitute another form taken by alternating-sign matrices; by assigning each monotone triangle a suitable weight, we can count domino tilings of an Aztec diamond.

In this paper, we study the wreath product of one-class association schemes K n = H (1, n) for n ≥ 2. We show that the d-class association scheme K n 1 K n 2 · · · K n d formed by taking the wreath product of K n i (for n i ≥ 2) has the triple-regularity property. Then based on this fact, we determine the structure of the Terwilliger algebra of K n 1 K n 2 · · · K n d by studying its irreducible modules. In particular, we show that every non-primary module of this algebra is 1-dimensional.

Understanding information transfer in the brain is a major challenge in today's neurosciences. Commonly, information transfer between brain areas is analyzed with the help of correlation measures for electrophysiological data. However, such approaches cannot distinguish between mutual coupling and other mechanisms of creating correlations between responses, such as common input from other sources. Functional coupling is mandatory for information transfer. Here we propose to analyze coupling between active brain areas with the help of models described by a system of differential-algebraic equations. Comparing models with various degrees of coupling, we show that mutual information transfer can be distinguished from one-way information transfer for activated cortical areas estimated by source localization techniques. We exemplify the technique with fast oscillatory activity found in both cortical areas 3b and 1 after peripheral nerve stimulation.

Several “classical” results on algebraic complete lattices extend to algebraic posets and, more generally, to so called compactly generated posets; but, of course, there may arise difficulties in the absence of certain joins or meets. For example, the property of weak atomicity turns out to be valid in all Dedekind complete compactly generated posets, but not in arbitrary algebraic posets. The compactly generated posets are, up to isomorphism, the inductive centralized systems, where a system of sets is called centralized if it contains all point closures. A similar representation theorem holds for algebraic posets; it is known that every algebraic poset is isomorphic to the system i(Q) of all directed lower sets in some poset Q; we show that only those posets P which satisfy the ascending chain condition are isomorphic to their own “up-completion” i(P). We also touch upon a few structural aspects such as the formation of direct sums, products and substructures. The note concludes w...

We analize the response of a single anharmonic diatomic molecule to a monochromatic time dependent electric field E ω (t) and evaluate the temporal evolution of the dipole moment, phase space trajectories and several transition probabilities as a function of the intensity of the field. We define deformed boson operators A † = f (n)a † , A = af (n) in terms of the usual harmonic oscillator creation and annihilation operators a, a † ,n = a † a and choose the number operator function f (n) such that the energy spectrum of a harmonic oscillator-type Hamiltonian written in terms of the deformed operators yield an energy spectra similar to that of a Morse potential.

Cryptographic identification schemes allow a remote user to prove his/her identity to a verifier who holds some public information of the user, such as the user public key or identity. Most of the existing cryptographic identification schemes are based on number-theoretic hard problems such as Discrete Log and Factorization. This paper focuses on the design and analysis of identity based identification (IBI) schemes based on algebraic coding theory. We first revisit an existing code-based IBI scheme which is derived by combining the Courtois-Finiasz-Sendrier signature scheme and the Stern zero-knowledge identification scheme. Previous results have shown that this IBI scheme is secure under passive attacks. In this paper, we prove that the scheme in fact can resist active attacks. However, whether the scheme can be proven secure under concurrent attacks (the most powerful attacks against identification schemes) remains open. In addition, we show that it is difficult to apply the conventional OR-proof approach to this particular IBI scheme in order to obtain concurrent security. We then construct a special OR-proof variant of this scheme and prove that the resulting IBI scheme is secure under concurrent attacks.

The disturbance rejection, defined as the problem of designing control laws that ensure, where possible, exogenous disturbances that do not affect the output of the perturbed system, has been resolved by means of algebraic and geometric techniques. This is a steady linear case by means of the static feedback state. Modifications of the Smith form through the SPR0 substitutions are presented which guarantees infinite zeros of the linear system.