Explicit congruences (mod p) are proved for the class equations or the products of class equations corresponding to discriminants D = −8p,−3p, −12p in the theory of complex multiplication, where p is an odd prime. These congruences are used to give a new proof of a theorem of Ogg, which states that there are exactly 15 primes p for which all j-invariants of supersingular elliptic curves in char...

We study the properties of Eisenstein-Kronecker numbers, which are related to special values of Hecke L-function of imaginary quadratic fields. We prove that the generating function of these numbers is a reduced (normalized or canonical in some literature) theta function associated to the Poincaré bundle of an elliptic curve. We introduce general methods to study the algebraic and p-adic proper...

Self-pairings are a special subclass of pairings and have interesting applications in cryptographic schemes and protocols. In this paper, we explore the computation of the self-pairings on supersingular elliptic curves with embedding degree k = 3. We construct a novel self-pairing which has the same Miller loop as the Eta/Ate pairing. However, the proposed self-pairing has a simple final expone...

Let d ≥ 1 be fixed. Let F be a number field of degree d, and let E/F be an elliptic curve. Let E(F )tors be the torsion subgroup of E(F ). In 1996, Merel proved the uniform boundedness conjecture, i.e., there is a constant B(d), which depends on d but not on the chosen field F or on the curve E/F , such that the size of E(F )tors is bounded by B(d). Moreover, Merel gave a bound (exponential in ...

The current methods of predictive controllers are utilized for those processes in which the rate of output variations is not high. For such processes, therefore, stability can be achieved by implementing the constrained predictive controller or applying infinite prediction horizon. When the rate of the output growth is high (e.g. for unstable nonminimum phase process) the stabilization seems to...

In this paper, a model of self-organizing spiking neural networks is introduced and applied to mobile robot environment representation and path planning problem. A network of spike-response-model neurons with a recurrent architecture is used to create robot’s internal representation from surrounding environment. The overall activity of network simulates a self-organizing system with unsupervise...

In this paper, an effective sliding mode design is applied to chaos synchronization. The proposed controller can make the states of two identical modified Chua’s circuits globally asymptotically synchronized. Numerical results are provided to show the effectiveness and robustness of the proposed method. Keywords—Sliding mode, Chaos synchronization, Modified Chua’s circuit.

This Classifying Bird Sounds (chip notes) project’s purpose is to reduce the unwanted noise from recorded bird sound chip notes, design a scheme to detect differences and similarities between recorded chip notes, and classify bird sound chip notes. The technologies of determining the similarities of sound waves have been used in communication, sound engineering and wireless sound applications f...