mathematics

Clearly the statement above is evident to anyone with a grounding in mathematics, but out of curiosity, I tried to think of the underlying definition that allows this, and found I really just 'knew ...

A simple linear problem goes min c'x such that f_i(x)<= 0 and Ax=b x Suppose we make all constraints affine. Then Dx-e<=0 and Ax-b =0 We get the Langrangian function as c'x + λ'(Dx-e) +ν'(Ax-b) and since Ax-b is 0, we reduce L to c'x + λ'(Dx-e) The dual function g is inf L(x,λ) x Then I... Read more

Researchers around the world are racing to develop new quantum-based systems for sensing, communication, computing and control that have the promise of outperforming traditional systems. Creating stable, measurable, distinguishable quantum states—which would be the heart of any such system—is a daunting task. Quantum states possess unique properties that can be exploited to develop novel informat…

I am reading through Errett Bishops &quot;Foundations of Constructive Analysis&quot; because I got interested in constructive as well as intuitionistic mathematics, and have a confusion in regards to ...

On simplicial de Rham cohomology?: Johan Louis Dupont: Simplicial de Rham Cohomology and characteristic classes of flat bundles, Topology 15 3 (1976) 233–245 [doi:10.1016/0040-9383(76)90038-0] Johan Louis Dupont: A dual simplicial de Rham complex, in: Algebraic Topology, Rational Homotopy, Lecture Notes in Mathematics 1318 (1988) [doi:10.1007/BFb0077796] On regulators and characteristic classes o…

Raoul Bott (1923–2005) was one of the great 20th century topologists and geometers. Among his famous works, one should mention the Bott periodicity theorem (of importance in K-theory), studies in Morse theory (including the study of Bott–Morse functions), the Borel–Weil–Bott theorem in geometric representation theory, the study of fixed point (localization) formulas (the Atiyah–Bott fixed point t…

On the simplicial de Rham complex and equivariant de Rham cohomology:

As a mathematician in approximation theory, moving from linear to non-linear structures feels like stepping into the unknown. When investigating Korovkin-type theorems for monotone sublinear operators, we realized power series methods could provide the exact theoretical framework we needed.

∞-Lie theory (higher geometry) Background Smooth structure Higher groupoids Lie theory ∞-Lie groupoids ∞-Lie algebroids Formal Lie groupoids Cohomology Homotopy Related topics Examples -Lie groupoids -Lie groups -Lie algebroids -Lie algebras superalgebra and (synthetic ) supergeometry A super Lie algebra which is a polyvector extension of the super Poincaré Lie algebra (supersymmetry) in for supe…

symmetric monoidal (∞,1)-category of spectra Let and be algebraic theories. The category of -bimodels and their homomorphisms is the category of -models and homomorphisms in . An alternative description is that it is a co--model in . Each such bimodel determines and is determined by a pair of adjoint functors Composition of such adjoint pairs yields a functor The category has a unit object – it w…

A morphism of sites is, unsurprisingly, the appropriate sort of morphism between sites. It is defined exactly so as to induce a geometric morphism between toposes of sheaves (or, more generally, exact completions). Let and be sites. A functor is a morphism of sites if is covering-flat, and preserves covering families, i.e. for every covering of an object , the family is a covering of . If has fin…

The term torsion is used for different concepts in different fields: In algebra, the torsion subgroup of a group is the group of elements of finite order (meaning: elements such that there is such that (with factors in the product)); similarly in ring theory an element of a module over a ring is a torsion element if it is annihilated by a nonzero element of the ring. A module is torsion (resp. to…

Riehl and Verity proved in [RV16] that the strict 2-category of the walking adjunction, usually denoted $\mathrm{Adj}$, has the expected universal property as the homotopy coherent walking adjunction. ...

BackgroundComputational thinking is a key higher-order cognitive skill in adolescence, extending beyond computer science to mathematical problem solving and complex reasoning. Although mathematical metacognition has been associated with computational thinking, the mechanisms underlying this relationship remain unclear. Grounded in Flavell’s metacognitive theory and Wing’s framework, this study ex…

The prolific mathematician discusses the role culture plays in understanding and appreciating science

The mathematician and former NFL player on the benefits of having a broad background for young people who are interested in science

I stumbled upon A065377, the list of primes which can't be written in the form $p+k^2$ ($p$ prime and $k&gt;0$ being an integer), and it's $2, 5, 13, 31, 37, 61, 127, 379, 439, 571, 829, 991, 1549, ...

I'm studying algebraic geometry and have a question on Kahler differentials. Let $k$ be a commutative ring with unit and $A$ a $k$-algebra. I already have the intuition if we picture $A$ as some ring ...

The Nisnevich topology, also called the completely decomposed topology, is a certain Grothendieck topology on the category of schemes which is finer than the Zariski topology but coarser than the étale topology. It retains many desirable properties from both topologies: The Nisnevich cohomological dimension (and even the homotopy dimension) of a scheme is bounded by its Krull dimension (like Zari…

His revolutionary idea? Before “computer science” was even a field, Church invented the lambda calculus (λ-calculus)—an elegant, abstract system for expressing computation through pure mathematical functions. In 1936, he used it to prove that no universal algorithm could ever decide the truth of all mathematical statements, solving Hilbert’s famous Entscheidungsproblem in the negative. This becam…