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Comparing orthogonal calculus and calculus with Reality [article]

Niall Taggart
2022 arXiv   pre-print
We show that there exists a suitable C_2-fixed points functor from calculus with Reality to the orthogonal calculus of Weiss which recovers orthogonal calculus "up to a shift" in an analogous way with  ...  formula analogous to the left adjoint of the functor from calculus with Reality, see e.g., [Tag22, Proposition 5.5].  ...  ., [BO13, Proposition 8.3]. Proposition 4.1.2. Let be a non-negative integer. The adjoint pair * !  ... 
arXiv:2205.15660v1 fatcat:jwpa46todvb65edzmmaa46hwza

Recovering unitary calculus from calculus with reality [article]

Niall Taggart
2021 arXiv   pre-print
calculus.  ...  By analogy with complex K–theory and K–theory with reality, there are theories of unitary functor calculus and unitary functor calculus with reality, both of which are generalisations of Weiss' orthogonal  ...  The author is thankful to Alexander Campbell for a helpful discussion on the cofinality statement for enriched ends in Proposition 4.2.  ... 
arXiv:2010.05948v2 fatcat:42lrxrcrozcrlbwwhmavj2ko6u

Strong Commutativity of Unbounded Self-adjoint Operators on a Separable Hilbert space

2020 Mathematical Theory and Modeling  
This article shows that by suitably selecting two unbounded positive Self-adjoint operators with compact inverses we get a set of pointwise commuting self-adjoint operators that commute on common core.  ...  The unbounded Self-adjoint operators that strongly commute on a common dense subset of their domain commute pointwise.  ...  The basics of unbounded Self-adjoint operators and their functional calculus are found in [1, 3, 8] .  ... 
doi:10.7176/mtm/10-8-03 fatcat:5kb72tynsvaxhitmykcopfk5i4

Dual-context Modal Logic as Left Adjoint of Fitch-style Modal Logic

Yoshihiko Kakutani, Yuito Murase, Yuichi Nishiwaki
2019 Journal of Information Processing  
Moreover, the adjunction is refined with polarity and the adjoint calculus is extended to polarized logic.  ...  In order to show the statement, we embed both the two systems, which are refined with levels, into the adjoint calculus that equips an adjunction a priori.  ...  under a hypothesis.  ... 
doi:10.2197/ipsjjip.27.77 fatcat:k5zts37c35deffthvdtvrs3jhe

Algebra, Proof Theory and Applications for a Logic of Propositions, Actions and Adjoint Modal Operators

Roy Dyckhoff, Mehrnoosh Sadrzadeh, Julien Truffaut
2012 Electronical Notes in Theoretical Computer Science  
We develop a cut-free (and hence analytic) nested sequent calculus for a modal logic of actions and propositions.  ...  The actions act on propositions via a dynamic modality (the weakest precondition of program logics), whose left adjoint we refer to as 'update' (the strongest postcondition).  ...  The sequent calculus of actions and adjoint modalities enriches the calculus of [12] with the two classical operators of ∨ and ∧.  ... 
doi:10.1016/j.entcs.2012.08.011 fatcat:ngoifnzwdbdn3l6xibc2473kpu

Unitary Functor Calculus with Reality [article]

Niall Taggart
2021 arXiv   pre-print
We construct a calculus of functors in the spirit of orthogonal calculus, which is designed to study "functors with reality" such as the Real classifying space functor, BU_ℝ(-).  ...  The calculus produces a Taylor tower, the n-th layer of which is classified by a spectrum with an action of C_2 ⋉ U(n).  ...  Moreover, U(n) + ∧ J R 1 (0, −) is the image of Σ ∞ + U(n) under the derived left adjoint Ä(L ψ ).  ... 
arXiv:2004.14886v2 fatcat:32arvgprbzbbdhra4co7bmrly4

Classification of bicovariant differential calculi

S. Majid
1998 Journal of Geometry and Physics  
The corresponding calculus is constructed and has dimension dimV 2 .  ...  Keywords: one-form { di erential calculus { quantum tangent space { quantum group { non-commutative geometry { quantum double  ...  Thus, the quantum tangent space for the inner calculus in Proposition 2.5 can be viewed as the kernel under di erentiation for a suitable (right-handed) calculus on H, taken along the direction of ?  ... 
doi:10.1016/s0393-0440(97)00025-9 fatcat:npw57xjwjbaqlbto6nfnz74jvq

May I borrow your logic? (Transporting logical structures along maps)

Maura Cerioli, José Meseguer
1997 Theoretical Computer Science  
They only depend upon a few abstract properties that hold under very general conditions given a pair of categories linked by adjoint functors.  ...  Therefore, all functors in the diagram have right adjoints. Inst &t Proposition 7 ([24, Proposition 311).  ...  Proposition 17. Let RI : C + D be the right adjoint of UI : D + C, and let RZ : D + E be the right adjoint of Uz : E + D.  ... 
doi:10.1016/s0304-3975(96)00160-0 fatcat:5lf3npegrngkzoubxz5uiq574q

Homotopy in functor categories

Alex Heller
1982 Transactions of the American Mathematical Society  
T c, for various C, are connected to one another by a functor calculus analogous to the <8>, Horn calculus for modules over rings.  ...  The functor calculus and the several homotopy theories may be articulated in such a way as to define an analogous functor calculus on the homotopy categories.  ...  Proposition 2. 5 . 5 The 2-cofibrations in 9"c are closed under right infinite composition in 9 cx2. 2-fibrations are closed under left infinite composition. Proposition 4.1.  ... 
doi:10.1090/s0002-9947-1982-0656485-2 fatcat:2qalzi5bojbcteyttlltwltfne

Homotopy in Functor Categories

Alex Heller
1982 Transactions of the American Mathematical Society  
T c, for various C, are connected to one another by a functor calculus analogous to the <8>, Horn calculus for modules over rings.  ...  The functor calculus and the several homotopy theories may be articulated in such a way as to define an analogous functor calculus on the homotopy categories.  ...  Proposition 2. 5 . 5 The 2-cofibrations in 9"c are closed under right infinite composition in 9 cx2. 2-fibrations are closed under left infinite composition. Proposition 4.1.  ... 
doi:10.2307/1998955 fatcat:ioticp77nvdihfttmybklluf44

The localization of orthogonal calculus with respect to homology [article]

Niall Taggart
2021 arXiv   pre-print
We give a variety of applications including a reformulation of the Telescope Conjecture in terms of our local orthogonal calculus and a calculus version of Postnikov sections.  ...  Our results also apply when considering the orthogonal calculus for functors which take values in spectra.  ...  Local orthogonal calculus Orthogonal functors Denote by C the category Top * of based topological spaces or the category Sp of (orthogonal) spectra.  ... 
arXiv:2109.13806v2 fatcat:5bhnaodxtzdkfon32wahast7n4

Models of Deduction*

Kosta Dosen
2006 Synthese  
the axioms of the lambda calculus.  ...  In this context, deductions are represented by arrows in categories involved in a general adjoint situation.  ...  in the axioms of the lambda calculus.  ... 
doi:10.1007/s11229-004-6290-7 fatcat:iljtb3tp4rehzpdoenkzf62gk4

Classification of Bicovariant Differential Calculi [article]

S. Majid
1996 arXiv   pre-print
The corresponding calculus is constructed and has dimension dim V^2.  ...  Thus, the quantum tangent space for the inner calculus in Proposition 2.5 can be viewed as the kernel under differentiation for a suitable (right-handed) calculus on H, taken along the direction of  ...  To see that it is closed under the adjoint action of U (g) it suffices to see that it is closed under the action of each ξ ∈ g. This action is by commutator in U (g).  ... 
arXiv:q-alg/9608016v2 fatcat:iyrpjjtblreu3ixk3kf5n3nf6y

Necessary Condition for an Euler-Lagrange Equation on Time Scales

Monika Dryl, Delfim F. M. Torres
2014 Abstract and Applied Analysis  
New and interesting results for the discrete and quantum calculus are obtained as particular cases.  ...  The study of optimal control problems on arbitrary time scales is a subject under strong current research [6, 7] .  ...  However, in T = R we can present this equivalence (Proposition 24). (15) is equivalent to a second order delta differential equation Proposition 23.  ... 
doi:10.1155/2014/631281 fatcat:nyicm6szrbhblesjmldttpdjw4

Advances in Quantum and Braided Geometry [article]

S. Majid
1997 arXiv   pre-print
construction and moduli space of all bicovariant calculi by means of some detailed examples, including finite-difference and 2-jet cacluli on ^n and full details of the Casimir construction of the 4D calculus  ...  This is the algebra of elements of A * fixed under the quantum adjoint action.  ...  These then transform under right translation via the adjoint action. Likewise, at the level of '1-forms'.  ... 
arXiv:q-alg/9610003v2 fatcat:zsxo52myujgifk765mfl5qb7ii
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