## ct.category Theory – Consistency theorem for tetracategories, \$ n \$ -class categories

Is there a coherence theorem / conjecture for tetracategories (weak $$4$$-categories)?

Todd Trimble mentions in his notes on tetracategories that his collage definitions are essentially unambiguous because of Gordon-Power-Street's tricategorical coherence theorem, and that it has been more than a decade since.

Does the current state of knowledge have a conjecture / coherence theorem for tetracategories, or even all $$n$$-categories?

## Number Theory – Uniform residue limits greater than \$ {1,2, dots, p-1 } \$

Let $$p$$ to be a first and $$m 1$$.

Define the ($$p$$ residue of a residue) function

$$tag 1 gamma _ {(p, m)}: [1, pm] to {0, dots, p-1 }$$
$$tag 2 quad quad quad quad quad quad quad quad quad quad ; n mapsto [pm pmod n] pmod {p}$$

To define

For $$1 to -1$$ to define

As long as the proof is missing (or $$text {(3)}$$ is refuted), allow me to make a conjecture.

My work

This came from my investigations here.

## sp.spectral theory – Inclusion of the spectrum of two differential operators defined on \$L^2[-a,a]\$ and \$L^2[0, infty)\$

Let $$T$$ be the formal operator defined by $$Tu:= sum_{j=0}^{2n} a_jfrac{d^ju}{dx^j}$$ where $$a_j in mathbb{C}$$. Consider the differential operators $$T_a: D(T_a)subseteq L^2[-a,a] to L^2[-a,a]$$ and $$T_infty: D(T_infty)subseteq L^2[0,infty) to L^2[0,infty)$$ defined by
$$T_af:=Tf, T_bg:=Tg, f in D(T_a), g in D(T_infty),$$
where $$D(T_a):={ f in L^2[-a,a] : Tf in L^2[-a,a], f^{(j)}(-a)=f^{(j)}(a)=0 mbox{ for } 0 leq j leq n-1}$$
and
$$D(T_infty):={ f in L^2[0,infty) : Tf in L^2[0,infty) , f^{(j)}(0)=0 mbox{ for } 0 leq j leq n-1}.$$

Can we say that $$sigma(T_a) subseteq sigma(T_infty)$$?. I know that the inclusion is true if we take $$Tu:=u”$$ or $$Tu:=-u”-2u’$$, for example.

## Theory of Representation – Drinfeld Polynomial for the Yangian \$ Y ( mathfrak {sl} _2) \$

I'm looking for direct evidence that a weight representation of $$Y ( mathfrak {sl} _2)$$ is of finite dimension if its highest weight is determined by a Drinfeld polynomial.

The results were recorded in the article of Tarasov in 1984 and also in that of Drinfeld in 1988. I found a proof of the similar result for quantum algebra affine $$U_q ( widehat { mathfrak {sl}} _ 2)$$ in the parper of Chari and Pressley in 1991 [Quantum Affine Algebras, Comm. Math. Phys. 142]. My question is: is it possible to formulate evidence similar to that of Chari and Pressley for the Yangian? $$Y ( mathfrak {sl} _2)$$.

I understand that there is detailed evidence in Molev's book. But this proof uses the fact that a finite representation of the highest and irreducible weight is a tensor product of the evaluation representations. I want to find a proof based on a direct calculation.

All comments and suggestions are appreciated.

## Spectral Theory on \$ { Psi } ^ { perp} \$

I take a self-adjoint operator $$A$$ from a Hilbert space, $$Psi$$ is a proper vector of eigenvalue $$a$$, I limit Hilbert space to $${ Psi } ^ { perp}$$ and I need to work with $$(A-a) ^ {- 1}$$. Does anyone know any references to this topic, please?

## User behavior – When should we use the theory of reasoned action (TRA)?

According to the theory of reasoned action (TRA) [1, 2], the model aims to measure the behavior of an individual. The theory is applied in social psychology literature that defines the relationships between beliefs, attitudes, norms, intentions, and behavior. However, we do not know when and how we should use it.

For this reason, we ask the following question:

When and how to use TRA?

During our work developments [3], we tried to understand when and how to use the TRA among the measures of user behavior. We followed the two questions on ResearchGate and the answers here on UX StackExchange. But we still do not have a concrete solution.

[1] Paul, J., Modi, A. and Patel, J., 2016. Predicting the consumption of ecological products with the help of the theory of planned behavior and reasoned action. Journal of Retail Sales and Consumer Services, 29, pp.123-134.

[2] Alryalat, M.A.A., Rana, N.P. and Dwivedi, Y.K., 2015. The adoption by citizens of an e-government system: validation of the expanded theory of reasoned action (TRA). International Journal of eGovernment Research (IJEGR), 11 (4), p.1-23.

[3] Calisto, F.M., Ferreira, A., Nascimento, J.C. and Gonçalves, D., 2017, October. Towards a diagnostic annotation of medical images based on touch. In Proceedings of the 2017 ACM International Conference on Surfaces and Interactive Spaces (pp. 390-395). ACM.

## Operator theory – A question related to spectral decomposition

Assume that $$( mathcal {A}, Phi)$$ is a semi-finite von Neumann algebra with trace $$Phi$$, $$B$$ is a positive self-adjoint operator affiliated with $$mathcal {A}$$, $$X$$ is a positive operator with $$Phi (X) < infty$$ who commutes with $$B$$, and $$E_n$$"S are an increasing sequence of spectral projections of $$X$$ with $$Phi (E_n) < infty$$ and $$lim E_n = s (X)$$, or $$s (X)$$ denotes the support of $$X$$. Let $$BX = int_0 ^ infty lambda dP ( lambda)$$ to be the spectral resolution of $$BX$$. then $$BXE_n = int_0 ^ infty lambda dP_n ( lambda)$$ or $$I-P_n ( lambda) = (I-P ( lambda)) E_n$$. Why is it true that $$Phi (BXE_n) = int_0 ^ infty Phi (I-P_n ( lambda)) d lambda?$$

## Theory of complexity – How to show that the product of two binary numbers can not be determined in \$ AC ^ {0} \$?

Contribution $$x = x_ {0} … x_ {n-1}$$. To determine the xor over $$n$$-parts $$x_i$$ he
is enough to multiply the next two $$n ^ 2$$Binary bits:
$$a = 0 ^ {n-1} hspace {2mm} x_ {n-1} hspace {2mm} 0 ^ {n-1} hspace {2mm} x_ {n-2} hspace {2mm} 0 ^ {n-1} hspace {2mm} … hspace {2mm} 0 ^ {n-1} hspace {2mm} x_ {1} hspace {2mm} 0 ^ {n-1} hspace {2mm} x_ {0}$$

$$b = 0 ^ {n-1} hspace {3mm} 1 hspace {5mm} 0 ^ {n-1} hspace {5mm} 1 hspace {5mm} 0 ^ {n-1} hspace { 2mm} … hspace {2mm} 0 ^ {n-1} hspace {5mm} 1 hspace {5mm} 0 ^ {n-1} hspace {5mm} 1$$

The product of two binary numbers can be determined in $$AC ^ {1}$$?

## group theory gr. – resolution of auther cayley for the quintic equation

hello i want to get different numbers using cayley resolvent

and I put integers x1 = 1 x2 = 2 x3 = 3 x4 = 4 x5 = 5

when to swap the integers in

= (X1x2 + x2x3 + x3x4 + x4x5 + x5x1
-X1x3 -x3x5 -x5x2 -x2x4 -x4x1) ^ 2

I've had (1,25,81,121) four digits but I'm not sure
is this the right way to think such a resolute?

## lo.logic – Does this pure class theory about the ordinals and their relations raise problems with respect to its arithmetic validity?

The following theory is a class theory, where all classes are either classes of ordinals, or relations between classes of ordinals, that is, classes of ordered pairs of Kuratowski. However, the size of his universe is poorly inaccessible. Ordinals are defined as von Neumann ordinals. The theory is formalized in a logic of first order with equality and belonging.

extensionality: $$forall z (z in x leftrightarrow z in y) to x = y$$

Comprehension: if $$phi$$ is a formula in which the symbol $$“ x "$$ is not free, so all closures of: $$exists x forall y (y in x leftrightarrow exists z (y in z) land phi)$$; are axioms.

Ordinal matching: $$forall text {ordinals} alpha beta exists x ( { alpha, beta } in x)$$

To define: $$langle alpha beta rangle = { { alpha }, { alpha, beta }$$

Ordinal addition:: $$forall text {ordinal} alpha exists x ( alpha cup { alpha } in x)$$

Reports: $$forall text {ordinals} alpha beta exists x ( langle alpha, beta row in x)$$

elements: $$exists y (x in y) to ordinal (x) lor exists text {ordinals} alpha beta (x = langle alpha, beta row)$$

Cut: $$ORD text {is weakly inaccessible}$$

Or $$ORD$$ is the class of all element ordinals.

/ Definition of the completed theory.

Now, this theory can clearly define various arithmetic operations extended on elemental ordinals. This also proves transfinite induction on elemental elements. In a sense, this can be considered as an arithmetic extensible to the infinite world. Of course $$PA$$ is interpretable in the finite segment of this theory.

In that response, Nik Weaver in his response raised the concern of ZFC as being arithmetically unfounded.

My question is this: assuming this theory to be consistent, is the problem of its arithmetic calculation identical to that of ZFC?

The reason for this question is that it seems to me that the above theory is only a naive extension of numbers to the infinite world, it has no power axiom nor equivalent. We can say that this theory is in a way purely mathematical in the sense that it only concerns numbers and their relations. Would this raise the same kind of suspicion about the non-arithmetic calculated with ZFC.

My reasoning on this point is that, in general, when one raises the concern about the arithmetical inadequacy of a theory, especially if this theory is welcomed by mathematicians working in set theory and the foundations, there must be a technical or intuitive argument behind this suspicion, otherwise this suspicion would be unfounded. Suspicion should not depend solely on the strength of the theory in question. Otherwise, we would define no stronger theory than $$PA$$ based on such concerns.

According to Nik Weaver's answer, it seems to me that his concern is based on the fact that ZFC does not intuitively capture a clear concept. However, this theory is based on an intuitive concept generally similar to that which defines the arithmetic of finite sets. He extends it intuitively very clearly, the higher ordinals are defined precedents successively, and it does not generally seem to be so different from the intuitive foundations of arithmetic in the finite world. The question is therefore whether this theory is still the prey of the arguments on which the concerns about the arithmetical non-compliance of ZFC are based.