Algebraic

We continue the study of the family of planar regions dubbed Aztec diamonds in our earlier article and study the ways in which these regions can be tiled by dominoes. Two more proofs of the main formula are given. The first uses the representation theory of GL(n). The second is more combinatorial and produces a generating function that gives not only the number of domino tilings of the Aztec diamond of order n but also information about the orientation of the dominoes (vertical versus horizontal) and the accessibility of one tiling from another by means of local modifications. Lastly, we explore a connection between the combinatorial objects studied in this paper and the square-ice model studied by Lieb.

This paper presents the first results on AIDA/cube, algebraic and sidechannel attacks on variable number of rounds of all members of the KATAN family of block ciphers. Our cube attacks reach 60, 40 and 30 rounds of KATAN32, KATAN48 and KATAN64, respectively. In our algebraic attacks, we use SAT solvers as a tool to solve the quadratic equations representation of all KATAN ciphers. We introduced a novel pre-processing stage on the equations system before feeding it to the SAT solver. This way, we could break 79, 64 and 60 rounds of KATAN32, KATAN48, KATAN64, respectively. We show how to perform side channel attacks on the full 254-round KATAN32 with one-bit information leakage from the internal state by cube attacks. Finally, we show how to reduce the attack complexity by combining the cube attack with the algebraic attack to recover the full 80-bit key. Further contributions include new phenomena observed in cube, algebraic and side-channel attacks on the KATAN ciphers. For the cube attacks, we observed that the same maxterms suggested more than one cube equation, thus reducing the overall data and time complexities. For the algebraic attacks, a novel pre-processing step led to a speed up of the SAT solver program. For the side-channel attacks, 29 linearly independent cube equations were recovered after 40-round KATAN32. Finally, the combined algebraic and cube attack, a leakage of key bits after 71 rounds led to a speed up of the algebraic attack.

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.

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.

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.

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.

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.

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).

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]

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&#x27;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 ...

For each integer k ≥ 1, we define an algorithm which associates to a partition whose maximal value is at most k a certain subset of all partitions. In the case when we begin with a partition λ which is square-bounded, i.e. λ = (λ 1 ≥ · · · ≥ λ k ) with λ 1 = k and λ k = 1, applying the algorithm times gives rise to a set whose cardinality is either the Catalan number c −k+1 (the self dual case) or twice that Catalan number. The algorithm defines a tree and we study the propagation of the tree, which is not in the isomorphism class of the usual Catalan tree. The algorithm can also be modified to produce a two-parameter family of sets and the resulting cardinalities of the sets are the ballot numbers. Finally, we give a conjecture on the rank of a particular module for the ring of symmetric functions in 2 + m variables.

For a permutation ω ∈ S n , Leclerc and Zelevinsky [9] introduced a concept of ω-chamber weakly separated collection of subsets of {1, 2, . . . , n} and conjectured that all inclusion-wise maximal collections of this sort have the same cardinality (ω) + n + 1, where (ω) is the length of ω. We answer affirmatively this conjecture and present a generalization and additional results.

Passive expressions in Algol-like languages represent computations that read the state but do not modify it. The need for such read-only computations arises in programming logics as well as in concurrent programming. It is also a central facet in Reynolds’s Syntactic Control of Interference. Despite its importance and essentially basic character, capturing the notion of passivity in semantic models has proved to be difficult. In this paper, we provide a new model of passive expressions using an automata-theoretic framework recently proposed by the author. The central idea is that the store of a program is viewed as an abstract form of an automaton, with a representation of its states as well as state transitions. The framework allows us to combine the strengths of conventional state-based models and the more recent event-based models to synthesize new ”automata-based” models. Once this basic framework is set up, relational parametricity does the job of identifying passive computations.

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.