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?

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.

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?