We prove a multiplication theorem of a quantum Caldero-Chapoton map associated to valued quivers which extends the results in [8][6]. As an application, when Q is a valued quiver of finite type or rank 2, we obtain that the algebra A H |k|(Q) generated by all cluster characters (see Definition 1) is exactly the quantum cluster algebra E H |k|(Q) and various bases of the quantum cluster algebras...

Using a distortion measure for the states emerging in self-organizing maps (SOMs) we mathematically analyse a recently proposed high-dimensional map formation model for ocular dominance patterns. We calculate critical values of parameters for ocular dominance states to occur, and we determine how the pattern layout depends on these parameters. The analysis reveals an increase of ocular dominanc...

1. Lyapunov Exponents for Differential Equations 2. Abstract Theory of Lyapunov Exponents 3. Regularity of Lyapunov Exponents Associated with Differential Equations 4. Lyapunov Stability Theory 5. The Oseledets Decomposition 6. Dynamical Systems with Nonzero Lyapunov Exponents. Multiplicative Ergodic Theorem 7. Nonuniform Hyperbolicity. Regular Sets 8. Examples of Nonuniformly Hyperbolic System...

In this paper, we give estimates of character sums of Weil type. They are associated with polynomials and rational functions with coefficients in a quadratic extension k2 of a finite field k of q elements, additive and multiplicative characters of k2, and summed over the kernel of the norm map from k× 2 to k×. The bounds are of the magnitude O( √ q) with the implied constant depending on the de...

We introduce a new identification scheme based on the Gap Diffie-Hellman problem. Our identification scheme makes use of the fact that the computational Diffie-Hellman problem is hard in the additive group of points of an elliptic curve over a finite field, on the other hand, the decisional Diffie-Hellman problem is easy in the multiplicative group of the finite field mapped by a bilinear map. ...

We show that the multiplicative domain of a completely positive map yields a new class of quantum error correcting codes. In the case of a unital quantum channel, these are precisely the codes that do not require a measurement as part of the recovery process, the so-called unitarily correctable codes. Whereas in the arbitrary, not necessarily unital case they form a proper subset of unitarily c...

This paper discusses the instability of eleven nonlinear state space models that underly exponential smoothing. Hyndman et al. (2002) proposed a framework of 24 state space models for exponential smoothing, including the well-known simple exponential smoothing, Holt’s linear and Holt-Winters’ additive and multiplicative methods. This was extended to 30 models with Taylor’s (2003) damped multipl...

Examples of multiplication by neurons or neural circuits are scarce, although many computational models use this basic operation. The owl's auditory system computes interaural time (ITD) and level (ILD) differences to create a two-dimensional map of auditory space. Space-specific neurons are selective for combinations of ITD and ILD, which define, respectively, the horizontal and vertical dimen...

Let L=K1 U .. , U Kp.r;;;S3 be a tame link of Jl~ 1 components with group G=7r1(S3_L). Also, let H=G/G' be the abelianization of G; H is then the (multiplicative) free abelian group generated by the meridians t1, ••• , tp. of L, and its integral group ring ZH consists of polynomials (with integer coefficients) in t 1 ,. .. , tp.' til, ... , t;l. The homomorphism 6: ZH~Z (which has 6(t;)=1 for e...

We define a generic multiplication in quantised Schur algebras and thus obtain a new algebra structure in the Schur algebras. We prove that via a modified version of the map from quantum groups to quantised Schur algebras, defined in [1], a subalgebra of this new algebra is a quotient of the monoid algebra in Hall algebras studied in [10]. We also prove that the subalgebra of the new algebra gi...