## Can the notation for polynomial reduction, A ≤p B be reversed in computability theory?

I don’t know this is a proper question on this forum but I was reading about computability theory and I saw the reduction concept and its notation like this: $$A \le_pB$$. I just wanted to know is this notation can be reversed? that is, can I write this down like $$A\ge_p B$$ And still have a meaning? I searched a lot but this notation always been like the former and I got confused.

## Mathematical Notation for Solution

I’m trying to write the solution to one of my problems, but am having a difficult time writing it out mathematically. I have a matrix $$C$$ that is given by $$C = [A^{N-1}B,A^{N-2}B,…,AB,B]\in\mathbb{R}^{n\times Nm},$$ where $$N$$ is a fixed number, and $$A\in\mathbb{R}^{n\times n}$$, $$B\in\mathbb{R}^{n\times m}$$. For now, lets work in the case with $$N=4$$ and $$n=3,m=1$$. In this case, $$C = [A^3B,A^2B,AB,B]\in\mathbb{R}^{3\times 4}$$ The solution to my problem is a vector $$u\in\mathbb{R}^{Nm} = \mathbb{R}^4$$ in this case. Onto the problem:

Each component of the solution $$u$$, i.e. $$u_k$$, takes a certain value whenever the condition $$\|B^\intercal (A^\intercal)^k\|_{\infty} = 1$$ is met, and takes the value of zero otherwise. Sounds like a simple piecewise function, but the problem is that this certain value comes from another vector, $$\tilde{u}$$, which has $$n$$ components, i.e. $$\tilde{u}\in\mathbb{R}^n$$. Thus, I need to put different indices on $$u_k$$ and $$\tilde{u}_n$$ and they must go in order. Let me give an example:

Suppose $$\tilde{u} = [0.25,0.75]$$, and $$\|B^\intercal (A^\intercal)^k\|_{\infty} = 1$$ is met for $$k = 1,4$$. Then, $$u$$ should be given by $$u = [u_1,u_2,u_3,u_4] = [0.25,0,0,0.75]$$. How can I formalize this?

## n¹⁰ = Ω(2ⁿ) ?? and n¹⁰ = O(2ⁿ) ?? True or False with O being the Big-O notation

n¹⁰ = Ω(2ⁿ) ?? and n¹⁰ = O(2ⁿ) ?? True or False with O being the Big-O notation

PS:please forgive me for not trying , i don’t understand Algorithms at all but i need help.

## How to write vectors in abbreviated set notation?

I was wondering whether anyone knew how to write a vectors in abbreviated set notation to express the solutions to this question:

“Determine all values of x, y, z ∈ R such that (x, y, z) is perpendicular to both a = (1, 1, 1) and b = (−1, 1, 1).”

Letting n=[x , y, z], I figured out the two simultaneous equations (we have not covered cross product yet)

1. x + y + z = 0
2. -x + y + z = 0

However, the question wants us to express the answer in the form of {…|c ∈ R) which I am unsure how to do.

I understanding expressing the answer in the regular notation would be something like {(x, y , z) ∈ $$R^3$$ | x=0 and y=-z}.

Thank you very much for your help guys! Much appreciated 🙂

## Need help understanding notation.

Proposition: Let $$G=(V,E)$$ denote a $$(v,k;\lambda, \mu)$$ strongly-regular graph. Then $$k(k-\lambda-1)=\mu(v-k-1)$$.

The notation is quite overloaded for me. Since $$k$$ and $$\mu$$ are parameters, then what does $$k(k-\lambda-1)$$ and $$\mu(v-k-1)$$ mean? Sorry if the quality of the question is not par to the quality of this site.

## Interpretation of an asymptotic notation

Assume that we measure the complexity of an algorithm (for some problem) by two parameters $$n$$ and $$m$$ (where $$m \le n$$). What is the formal interpretation of the following claim: there is no algorithm that solves the given problem in $$o(m + \log{n})$$?

In particular, does it mean that an $$O(\log{n})$$ algorithm is possible?

## Babilonic notation to decimal notation. Example $1;12 \cdot 15$

I’m currently working in a program that convert numbers in babilonic notation into decimal numbers. The problem I have is that the example and requirements described by the teacher deliver numbers in the following format

$$1;12 \cdot 15$$

That would be a number on its “babilonic” structure. The result after some operations that I trully don’t know and were shown by the teacher really fast seems like $$72,25$$ in decimal notation.

That was the example provided and I’m not too clear about it. I’ve found something similar in Wikipedia referring to calculation of irrational numbers starting from a sexagesimal structure similar to the one provided but I’ve found is not the same.

I hope somebody has any information about babilonic numbers and the notation provided because further than Wikipedia I haven’t found something closer to my problem, any hint or help will be really appreciated.

## Differential equation notation about maximal solution

I’m doing the following problem: The differential equation $$\dot{y} = X(t,y), X(t,y) = \frac{1}{3}y^{1/4} +t^{1/3}$$ defined on $$D_X = (0,\infty)\times(0,\infty)$$.

I already solved it with: $$y(t) = t^{4/3}$$.

But here is what i don’t understand. The problem says:

For $$\eta >0$$ let $$(I_\eta,y_\eta)$$ denote the maximal solution with $$y_\eta(1) = \eta$$

for:

a) For $$0 < \eta < 1$$ Then $$y_\eta(t) < t^{4/3}$$, for $$t \in I_\eta$$

b) For $$\eta > 1$$ Then $$y_\eta(t) > t^{4/3}$$, for $$t \in I_\eta$$

I am very confused about the $$y_\eta(1) = \eta$$ notation, so i can’t understand what the goal with the task is. Can you help?

## How to present a statement in predicate notation

Let E (p, q) be the statement “p has emailed q” where the Universe o discourse for both p and q is the set of all students in class 2018. Use quantifiers to express the following statements. James has received email from exactly two persons in the class.

I found this question in a past exam paper. I totally have no idea what’s going on. I am new to this, help I just need some clues no answers

## Java Optional functional notation

Up to now, I’ve been able to write this code:

public class ReferenceResourceImpl implements ReferenceResource {      private final transient CacheControl cacheControl;     private final transient Request request;      private final transient ReferenceService referenceService;      public ReferenceResourceImpl(         @CacheControlConfig(maxAge=20) CacheControl cacheControl,         @Context Request request,         ReferenceService referenceService     ) {         this.cacheControl = cacheControl;         this.request = request;          this.referenceService = referenceService;     }      /**      * Calculates {@link Reference}'s {@link EntityTag}.      * @param reference {@link Reference} to look up      * @return Calculated {@link EntityTag}      */     private EntityTag eTag(Reference reference) {         return EntityTag.valueOf(Integer.toString(reference.hashCode()));     }      /**      * {@inheritDoc}      */     @Override     public Response download(         String id     ) {         LOG.info(RestConstants.Logs.UPLOAD_START, id);          Optional<Reference> reference = this.referenceService.get(id);          ResponseBuilder responseBuilder = this.request.evaluatePreconditions(this.eTag(reference.get()));         if (Objects.isNull(responseBuilder)) {             responseBuilder = Response                 .ok(reference.get())                 .cacheControl(this.cacheControl)                 .tag(this.eTag(reference.get()));         }          return responseBuilder.build();     } } 

I’d like to refactor this code using functional “notation”:

Optional<Reference> reference = this.referenceService.get(id);  ResponseBuilder responseBuilder = this.request.evaluatePreconditions(this.eTag(reference.get())); if (Objects.isNull(responseBuilder)) {     responseBuilder = Response         .ok(reference.get())         .cacheControl(this.cacheControl)         .tag(this.eTag(reference.get())); }  return responseBuilder.build(); 

I’ve tried that code:

return this.referenceService.get(id)   <<<< Fetch entity from database     .map(this::eTag)     .map(this.request::evaluatePreconditions)     .orElse(Response.status(Status.NOT_FOUND))     .cacheControl(this.cacheControl)     .tag(this.eTag(this.referenceService.get(id).get()))  <<<< Fetch entity from database     .build(); 

Here I find code more elegant, but I fetching twice my entity on database using this.referenceService.get(id).

Any ideas about how to solve that?