C.H. Yang discovered a polynomial version of the classical Lagrange identity expressing the product of two sums of four squares as another sum of four squares. He used it to give short proofs of some important theorems on composition of δ-codes (now known as T -sequences). We investigate the possible new versions of his polynomial Lagrange identity. Our main result shows that all such identitie...

Abstract. Let Jμ[X; q, t] be the integral form of the Macdonald polynomial and set H̃μ[X; q, t] = tJμ[X/(1− 1/t); q, 1/t ], where n(μ) = ∑ i(i− 1)μi. This paper focusses on the linear operator ∇ defined by setting ∇H̃μ = tq ′)H̃μ. This operator occurs naturally in the study of the Garsia-Haiman modules Mμ. It was originally introduced by the first two authors to give elegant expressions to Frobeni...

In the olfactory system, odor percepts retain their identity despite substantial variations in concentration, timing, and background. We propose a novel strategy for encoding intensity-invariant stimuli identity that is based on representing relative rather than absolute values of the stimulus features. Because, in this scheme, stimulus identity depends on relative amplitudes of stimulus featur...

We use the Lagrange identity method and the logarithmic convexity to obtain uniqueness and exponential growth of solutions in the thermoelasticity of type III and thermoelasticity without energy dissipation. As this is not the first contribution of this kind in this theory, it is worth remarking that the assumptions we use here are different from those used in other previous contributions. We a...

Recently, a few biometric identity-based encryption (BIO-IBE) schemes have been proposed. BIO-IBE leverages both fuzzy extractor and Lagrange polynomial to extract biometric feature as a user public key and as a preventive measure of collusion attack, respectively. In this paper, we reveal that BIO-IBE is not realistic whereby a query of fresh biometrics is needed for each encryption process. M...

In the following we discuss a well-known binomial identity. Many proofs by different methods are known for this identity. Here we present another proof, which uses linear ordinary differential equations of the first order. Several proofs of the well-known identity n ∑ k=0 ( n + k n ) 2 = 2 (1) [4, (1.79)] appear in the literature. In [3, Equation (5.20)], it is proved using partial sums of bino...

We present a new operator equality in the framework of Hilbert C∗-modules. As a consequence, we get an extension of the Euler–Lagrange type identity in the setting of Hilbert bundles as well as several generalized operator Bohr’s inequalities due to O. Hirzallah, W.-S. Cheung–J.E. Pečarić and F. Zhang.