Ciprian Demeter coauthored the decoupling inequality with Jean Bourgain. It bounds global Fourier contributions by a quadratic sum of contributions from small caps near a curved hypersurface. The theorem turns geometric curvature into an analytic inequality and yields consequences for Strichartz estimate for the free Schrödinger equation and discrete restriction problems.^[ref]

Jean Bourgain proved the decoupling inequality with Ciprian Demeter. Their theorem controls a function with Fourier support near a curved hypersurface through contributions from smaller frequency caps. The result connects curvature with improved summation estimates, with applications to restriction and Strichartz estimate for the free Schrödinger equation.^[ref]

Russ Tedrake authored Underactuated Robotics by Russ Tedrake, developing mathematical methods for controlling systems with fewer independent actuators than degrees of freedom. Its treatment of the linear-quadratic optimal control derives feedback gains from a Riccati equation. This connects a local dynamical model and a quadratic performance objective with an explicit stabilizing control law under appropriate assumptions.^[ref]

Werner Dinkelbach developed a method for nonlinear fractional programming. Instead of directly optimizing a ratio $f(x)/g(x)$ with positive denominator, the method repeatedly optimizes $f(x)-\lambda g(x)$ and updates $\lambda$ from the resulting ratio. The transformation connects the optimal ratio with a zero of a scalar value function, giving the ratio problem a sequence of simpler parametric subproblems.^[ref]

Niels Henrik Abel proved that the general quintic equation does not have solvability by radicals. His 1824 pamphlet established a limitation on formulas built from arithmetic operations and repeated extraction of roots, rather than a failure of numerical solution. Particular quintic equations can still have solvability by radicals; the impossibility concerns a formula for the general case.^[ref]

The result, expressed in the Abel-Ruffini theorem, helped shift the study of polynomial equations from searching for ever larger formulas toward asking which structural properties permit an algebraic solution.

Lowell Schoenfeld coauthored quantitative estimates in analytic number theory with J. Barkley Rosser. Their paper gives explicit bounds for functions of prime numbers, including an upper bound for $n/\phi (n)$. Careful constants, exceptional cases and validity ranges distinguish such a result from a purely asymptotic statement.^[ref]

J. Barkley Rosser coauthored explicit estimates for prime-related functions with Lowell Schoenfeld. Their bounds include the Rosser–Schoenfeld totient bound, controlling $n/\phi (n)$ through logarithmic expressions with stated numerical constants and ranges. Explicit inequalities make number-theoretic estimates usable at finite values, where an asymptotic equivalence alone may give insufficient error control.^[ref]

Seth Viren Neel authored notes on binary quadratic forms and the ideal class group. The notes develop equivalence, reduction and genus theory, showing how bounded coefficients permit finite enumeration of reduced positive definite forms. This gives a concrete computational view of class numbers while retaining the distinction between a form and its equivalence class.^[ref]

Robert H. Litzenberger studied state-contingent claim prices with Douglas T. Breeden. Their call-spread argument relates derivatives of call option prices with respect to strike to state prices. Under suitable smoothness, the second derivative yields a discounted density, providing a bridge between observable option pricing and the pricing of narrow ranges of future asset values.^[ref]

Douglas T. Breeden coauthored the extraction of state-contingent prices from option pricing with Robert H. Litzenberger. For sufficiently regular call prices across strikes, their second strike derivative identifies a discounted state-price density. The construction makes an entire option-price curve informative about the market's pricing of future outcomes, rather than only one payoff.^[ref]

Josef Teichmann authored lectures on the fundamental theorem of asset pricing. The notes connect no-arbitrage, equivalent local martingale measures and replication, including a finite-state convex-geometric argument. This explains why a collection of quoted prices can be consistent precisely when an appropriate positive pricing measure exists under the theorem's hypotheses.^[ref]

Ander Murua coauthored complex-coefficient operator splitting methods with Sergio Blanes and Fernando Casas. They analyze how composition coefficients influence order and error propagation. Positive real parts allow complex-time substeps to bypass an obstruction associated with negative real coefficients, particularly relevant to dissipative evolution equations.^[ref]

Fernando Casas studied operator splitting with Sergio Blanes and Ander Murua. Their work constructs higher-order compositions using complex coefficients with positive real parts. The approach addresses the negative-step obstruction of real-coefficient splittings, making high-order integration possible in settings where backward evolution, such as backward diffusion, is problematic.^[ref]

Sergio Blanes coauthored methods for splitting differential-equation evolution with Fernando Casas and Ander Murua. They studied complex composition coefficients with positive real parts, avoiding the backward real-time steps required by higher-order real splittings. This matters for parabolic partial differential equations, where a backward diffusion substep can be ill posed.^[ref]

Piotr Hajłasz authored notes on functional analysis developing Banach spaces, Hilbert spaces and the Baire category theorem. One application proves that a smooth real function is a polynomial if, at every point, at least one of its derivatives vanishes. The proof illustrates how a countable union of local conditions can force a global algebraic conclusion.^[ref]

Sasha Sodin studied the Erdős-Turán inequality and quantitative errors in Wigner semicircle distribution with Ohad N. Feldheim. Their work connects estimates on Fourier or polynomial moments with discrepancies between probability measures. Such bounds make convergence of global spectral distributions quantitative, rather than merely asserting a limiting law.^[ref]

Ohad N. Feldheim coauthored a proof of the Erdős-Turán inequality and an error estimate for Wigner semicircle distribution with Sasha Sodin. The discrepancy bound controls deviations of a measure on the unit circle through finitely many Fourier coefficients. It converts harmonic information into a quantitative statement about spatial distribution.^[ref]

Bo Åhlander coauthored a study of the arithmetic derivative with Victor Ufnarovski. The operation is determined on positive integers by assigning one to each prime number and imposing the product rule. Thus the derivative of $6=2\cdot 3$ is $3+2=5$, illustrating an arithmetic analogue of differentiation built from integer factorization.^[ref]

Victor Ufnarovski studied the arithmetic derivative with Bo Åhlander. Their definition assigns derivative one to every prime number and extends by the product rule. For $n=\prod p_i^{a_i}$, the resulting formula $n{\sum }_i{a}_i/p_i$ turns differentiation into an operation governed by integer factorization, rather than an ordinary derivative of a real-variable function.^[ref]

Neal Parikh coauthored an exposition of the proximal point algorithm with Stephen Boyd. A proximal operator replaces a difficult nonsmooth minimization step with a quadratic-regularized subproblem. Their account connects this operation with convex projection, resolvents and splitting methods, allowing nonsmooth objectives and constraints to be treated within a common algorithmic framework.^[ref]