Solid-Phase Synthesis of a Biotin Derivative and its Application to the Development of Anti-Biotin Antibodies

A biotin derivative, namely biotin–aminocaproic acid–lysine (BAL), was synthesized with solid-phase chemistry, conjugated to a carrier-protein, and used for rabbit immunization. The aminocaproic acid–lysine “long-arm” was used in order to project the biotin-hapten above the carrier-protein surface. Lysine was selected due to its N^ε-amino group, through which BAL was conjugated to the carrier-protein. BAL was synthesized on a commercially available resin with the Fmoc-solid-phase strategy; this has simplified the experimental procedure, overcome the need for intermediate purification steps, and led to a final product of high purity, with high yield. The anti-BAL antibodies recognized free biotin, as shown with an in-house-developed ELISA, in which biotin conjugated to a synthetic “lysine–dendrimer” was used to coat the ELISA microwells. In immunocytology and Western-blot experiments, the anti-BAL antibodies led to similar results with those obtained with streptavidin. Synthetic derivatives of hapten molecules that can be easily prepared with solid-phase chemistry, such as BAL, may be used for the development of specific antibodies for the corresponding hapten.     

What is your “birthday elliptic curve”?

In this article, Ramanujan–Weber class invariants and its analogue are used to derive birthday elliptic curves.     

‘Universal’ inequalities for the eigenvalues of the Hodge de Rham Laplacian

We obtain ‘universal’ inequalities for the eigenvalues of the Laplacian acting on differential forms of a Euclidean compact submanifold. These inequalities generalize the Yang inequality concerning the eigenvalues of the Dirichlet Laplacian of a bounded Euclidean domain.      

Second Eigenvalue of Schrödinger Operators¶and Mean Curvature

Let $M$ be a compact immersed submanifold of the Euclidean space, the hyperbolic space or the standard sphere. For any continuous potential q on M, we give a sharp upper bound for the second eigenvalue of the operator −Δ+q in terms of the total mean curvature of M and the mean value of q. Moreover, we analyze the case where this bound is achieved. As a consequence of this result we obtain an alternative proof for the Alikakos–Fusco conjecture concerning the stability of the interface in the Allen–Cahn reaction diffusion model. Received: 18 June 1999 / Accepted: 6 July 1999     

Free Energy Wells and Overlap Gap Property in Sparse PCA

We study a variant of the sparse PCA (principal component analysis) problem in the “hard” regime, where the inference task is possible yet no polynomial-time algorithm is known to exist. Prior work, based on the low-degree likelihood ratio, has conjectured a precise expression for the best possible (subexponential) runtime throughout the hard regime. Following instead a statistical physics-inspired point of view, we show bounds on the depth of free energy wells for various Gibbs measures naturally associated to the problem. These free energy wells imply hitting time lower bounds that corroborate the low-degree conjecture: we show that a class of natural MCMC (Markov chain Monte Carlo) methods (with worst-case initialization) cannot solve sparse PCA with less than the conjectured runtime. These lower bounds apply to a wide range of values for two tuning parameters: temperature and sparsity misparametrization. Finally, we prove that the overlap gap property (OGP), a structural property that implies failure of certain local search algorithms, holds in a significant part of the hard regime. © 2022 Wiley Periodicals, Inc.     

A Reilly inequality for the first Steklov eigenvalue

Let M be a compact submanifold with boundary of a Euclidean space or a Sphere. In this paper, we derive an upper bound for the first non-zero eigenvalue p₁ of Steklov problem on M in terms of the r-th mean curvatures of its boundary ∂M. The upper bound obtained is sharp.     

Universal inequalities for the eigenvalues of a power of the Laplace operator

In this paper, we obtain a new abstract formula relating eigenvalues of a self-adjoint operator to two families of symmetric and skew-symmetric operators and their commutators. This formula generalizes earlier ones obtained by Harrell, Stubbe, Hook, Ashbaugh, Hermi, Levitin and Parnovski. We also show how one can use this abstract formulation both for giving different and simpler proofs for all the known results obtained for the eigenvalues of a power of the Laplace operator (i.e. the Dirichlet Laplacian, the clamped plate problem for the bilaplacian and more generally for the polyharmonic problem on a bounded Euclidean domain) and to obtain new ones. In a last paragraph, we derive new bounds for eigenvalues of any power of the Kohn Laplacian on the Heisenberg group.     

Laplacian eigenvalue functionals and metric deformations on compact manifolds

In this paper, we investigate critical points of the eigenvalues of the Laplace operator considered as functionals on the space of Riemannian metrics or a conformal class of metrics on a compact manifold. We introduce a natural notion of the critical metric of such a functional and obtain necessary and sufficient conditions for a metric to be critical. We derive specific consequences concerning possible locally maximizing metrics. We also characterize critical metrics of the ratio of two consecutive eigenvalues.