For example, if someone sees someone on the street, they may have an opinion about how old that person is. That conjecture may or may not be true. The person could find out for sure if their conjecture is true or false by asking the person their age. In math, the definition of a ...
Here, Imψ denotes the imaginary part of a function, in distinction to the meaning of Im above for the imaginary part of an operator. Let N:={x∈U:ψ(x)=0} be the nodal set of ψ, and set V:=−‖K‖−11U∖NImψψ. Note that the nodal set is well defined since ϕ ...
We're one step closer to solving an 82-year-old math riddle that has stumped the world’s smartest mathematicians. But a complete answer could still be far away.
2.1.4 Gauge-singlet constraint Let us discuss the meaning of the ungauging in terms of quantum states in the Hilbert space. By using the gauge-invariant Hilbert space Hinv and the extended Hilbert space Hext, thermal partition functions are written as (see e.g., appendix A.2 in ref. [...
For example, the claim that the probability that a particular individual atom will decay after time interval t may perhaps more reasonably be thought of as meaning that the probability refers to the relative frequency of decay after time t in a set of atoms. The reference to a larger ...
The coefficient ak has a combinatorial meaning: it is the number of degree k effective divisors over X. Using the relation between line bundles and divisors, we have the following formula ak=∑L∈Pick(X)qh0(X,L)−1q−1. Indeed, for any divisor D, we have Γ(X,L(D))={f∈K(...
Let’s unpack the automorphic side of the geometric Langlands conjecture as presented in the paper, specifically the notation "D-mod1/2(BunG)." This term is dense with meaning, so I’ll break it down step-by-step to give you a clear picture of what’s going on, while keeping it natu...
the Clay Institute announced it would offer a one-million-dollar reward apiece for solutions to the problems. While a layperson might have a tough time penetrating the quantum physics behind the Yang-Mills and Mass Gap problem, they have no difficulty understanding the meaning of the number 1 ...
The element Pγ has geometric meaning similar to that of Pr . It describes the cohomology class of a degeneracy locus r (γ ) defined in [4]. If we are given a tableau diagram, the notion of a factor sequence can also be extended to paths. Any factor sequence for a path γ will ...
, meaning that its large-scale behaviour is the same as it would be in a ‘geometrically trivial’ setting such as the complete graph or the binary tree. for many models the mean-field regime is described by alexander–orbach behaviour [ 5 , 14 , 46 ], in which the relevant random ...