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In mathematics, a '''weak trace class''' operator is a [[compact operator]] on a [[separable space|separab ...is the smallest two-sided ideal for which all traces on it are [[singular trace]]s. ...

...c [[modular form]]. It is a specialization of the more general [[Kuznetsov trace formula]]. In its simplest form the Petersson trace formula is as follows. Let <math>\mathcal{F}</math> be an [[orthonormal bas ...

...=Krupnik|title=Traces and Determinants of Linear Operators|series=Operator Theory Advances and Applications |publisher=Birkhäuser |place=Basel |date=1991 |is The theorem should not be confused with the [[Grothendieck trace formula]] from [[algebraic geometry]]. ...

...ry]], the '''Kuznetsov trace formula''' is an extension of the [[Petersson trace formula]]. ...'' formula connects [[Kloosterman sum]]s at a deep level with the spectral theory of [[automorphic form]]s. Originally this could have been stated as follows ...

...Kleinian]] and [[Fuchsian group]]s, though related objects are used in the theory of [[Lattice (discrete subgroup)|lattices]] in Lie groups, often under the === Trace field and invariant trace fields for Fuchsian groups === ...

...''categorical trace''' is a generalization of the [[trace (linear algebra)|trace]] of a [[matrix (mathematics)|matrix]]. ...</math> playing the role of a dual object of ''X''. In this situation, the trace of a [[morphism]] <math>f: X \to X</math> is defined as the composition of ...

{{Short description|Algebraic trace}} ...e of [[trace class operator]]s. Dixmier traces are examples of [[singular trace]]s. ...

In the [[mathematics|mathematical]] theory of [[Lie group]]s, the '''Chevalley restriction theorem''' describes functi ...ak{sl}_n</math>, the Lie algebra of matrices with [[trace (linear algebra)|trace]] zero ...

...[[quantum information]] and the study of [[open quantum system]]s, the '''trace distance''' is a [[metric (mathematics)|metric]] on the space of [[density ...ath>A</math> taking the algebraic square roots of its eigenvalues. For the trace distance, we more specifically have an expression of the form <math>|C|\equ ...

In [[dynamical systems theory]], the '''Olech theorem''' establishes [[Necessity and sufficiency|sufficie :(a) the [[Trace (linear algebra)|trace]] of the [[Jacobian matrix and determinant|Jacobian matrix]] is negative, < ...

...n inequality in [[information theory]] relating the [[Entropy (information theory)|entropy]] of a set of variables to the entropies of a collection of subset ...le=Some Intersection Theorems for Ordered Sets and Graphs|journal=J. Comb. Theory A|date=1986|volume=43|pages=23–37|doi=10.1016/0097-3165(86)90019-1|doi-acce ...

In the mathematical theory of [[free probability]], the notion of '''free independence''' was introduc In the context of Voiculescu's free probability theory, many classical-probability theorems or phenomena have free probability ana ...

...<math>\mathcal{M}</math> be a finite factor with the canonical normalized trace <math>\tau</math> and let <math>X</math> be an invertible operator in <math ...rminant#Relation to eigenvalues and trace|Relation between determinant and trace via eigenvalues]]). The number <math>\Delta(X)</math> is well-defined by [[ ...

...values of ''s''. Here "tr" denotes a functional [[trace (linear algebra)|trace]]. One of the most important motivations for [[Arakelov theory]] is the zeta functions for operators with the method of [[heat kernel]]s g ...

...edia article [[seismic inverse Q filtering]]. The solution is based on the theory of wavefield downward continuation. In this outline here I will compute on ...trace is scaled individually, so artefacts are stronger on trace 5 than on trace 4). However, Wang also considered phase compensation. Computations here are ...

...C) |pages=20–30 |year=1997}}</ref><ref name="watrous_book">J. Watrous. The Theory of Quantum Information. Cambridge University Press, 2018. Chapter 3.3, http ...trivial extension of a linear map, maximized over all possible inputs with trace norm at most one. More precisely, let <math>\Phi: M_n(\mathbb{C}) \to M_m(\ ...

...pies]] of two [[density matrix|density matrices]] as a function of their [[trace distance]]. It was proved by Koenraad M. R. Audenaert in 2007<ref>Koenraad is the trace distance between the two matrices. Note that the base for the [[logarithm] ...

...he '''Arthur–Selberg trace formula''' is a generalization of the [[Selberg trace formula]] from the group SL₂ to arbitrary [[reductive group]]s o ...rv|Flicker|Kazhdan|1988}} is less general but easier to prove. The [[local trace formula]] is an analogue over local fields. ...

...|Tate|1968}} proves the theorem using a notion of [[trace (linear algebra)|trace]]s for certain [[endomorphism]]s of [[dimension (vector space)|infinite-dim ...len|last2=Kleiman|first2=Steven|title=Introduction to Grothendieck duality theory|series=Lecture Notes in Mathematics|volume=146|publisher=Springer|year=1970 ...

...s. Furthermore, let ''τ'' = ''A'' + ''D'' be the [[Trace (linear algebra)|trace]] of ''M'', and ''δ'' = ''AD'' − ''BC'' be its [[determinant]]. Let ''s'' If the determinant ''δ'' is zero, but the trace ''τ'' is nonzero, the general formula above will give only two distinct sol ...