Functional analysis: are there no true functions with real values ​​of two or more variables?

I see the following theorem somewhere, without complete proof.

For each irrational number $ lambda $, there are continuous functions $ phi_k: (0,1) to mathbb R, k = 1,2,3,4,5 $, so that for all continuous functions $ f: (0,1) ^ 2 to mathbb R $, there is a continuous function $ g: (0,1) a mathbb R $such that $ f (x, y) = sum_ {k = 1} ^ 5 g ( phi_k (x) + lambda phi_k (y)) $.

There are some mysterious things about this result. Where does the number come from? $ 5 $ come and why $ lambda notin mathbb Q $?

Could anyone offer a reference for proof of this?

special functions: integral expressions for Bessel type power series

I am interested in power series form $$ f (z) = sum_ {k = 0} ^ infty frac {z ^ k} {(k!) ^ alpha}. $$ When $ alpha = 1 $, this becomes $ exp (z) $. by $ alpha = 2 $ this is a function of Bessel and for larger integers $ alpha $ We obtain a hypergeometric series. These special functions ($ alpha> 1 $) have integral expressions in the form of some integral of an elementrary function.

Can you say something for non-integral values ​​of $ alpha $? Yes $ alpha> 1 $ it is an arbitrary real number, is there any hope of writing an integral expression for the sum? In general, has this form of series been studied somewhere in the literature? Any useful techniques to work with them?

Epsilon Delta test of rational functions

I'm stuck trying this problem then how do I proceed?
Solution

How do I execute multiple onEdit functions on the same spreadsheet in Google Sheets?

everybody!
I look forward to having your experience to help me with a project I am working on.

I want to call different functions based on which spreadsheet the event occurred.
Ex. If you type 1 in B6 in "Room reservation", checkB6 () will be called. If you type 1 in E17 for "Supplies", checkE17 () will be called.

Everything seemed to be working yesterday, and now it's not like that, although I didn't change anything to onEdit (e).

Here is my code:


//When something is entered into the target cell, the appropriate function is called automatically
function onEdit(e){

var s = SpreadsheetApp.getActiveSpreadsheet().getActiveSheet().getName();

if (s = "Room Requests") {
     e = SpreadsheetApp.getActiveSpreadsheet().getSheetByName("Room Requests").getRange("b8").getValue();
     checkB8();
} 



else if (s = "Car Requests") {
         e = SpreadsheetApp.getActiveSpreadsheet().getSheetByName("Car Requests").getRange("b9").getValue();
         checkB9();
} 



else if (s = "Supplies") {
         e = SpreadsheetApp.getActiveSpreadsheet().getSheetByName("Supplies").getRange("e17").getValue();
         checkE17();
} 

  //etc., etc., etc.

else {e = SpreadsheetApp.getActiveSpreadsheet().getSheetByName("Birthdays").getRange("b4").getValue();
   checkB4();
 }
}

algorithms – Question about asymptotic analysis comparing two functions

I would like to receive an explanation about the analysis of this exercise. Given these functions: $$ f (n) = n ^ 2 \ g (n) = n ^ {2/3} $$

Show that $ f (n) = O (g (n)) $or $ f (n) = Omega (g (n)) $ and comment if $ f (n) = Theta (g (n)) $

In my answer, it is $ Theta (g (n)) $ with $ n_0 = 1, C = 1 $

P.S: The exercise requires a mathematical demonstration of the answer.

3.2 Hmwk – Higher-grade polynomial functions

Use a graphical utility to graph the function.
f (x) = 1 / 2xˆ4 – 7xˆ2
Use the zero or root characteristic to approximate the real zeros of the function. Then determine if the multiplicity of each zero is even or odd. (Round each zero to one decimal place).

Applying functions to members of members … of lists of lists …, etc.

Suppose I have a depth list $ k $, and I want to apply a function $ f $ to all the level elements $ n $ – e.g. $ f, { { {a, b }, {b, c } }, { {c, d } } }, 3 mapsto { { {f ( a), f (b) }, {f (b), f (c) } }, { {f (c), f (d) } } } $.

Presumably this can be done with some combination of the integrated elements. Map, Level, ApplyY Depth, but I can't seem to make it work. Is there any way to define a function? Function(,,) Which one applies a function only to terms with some depth in a list of lists?

MATLink Parallelization: call user-defined MATLAB functions

I am trying to run parallel simulations of finite elements in Mathematica using the AceGen and AceFEM software along with a MATLAB script called from Mathematica.

I can run some of the simulations that do not require the MATLAB script in parallel using the pseudo approach below.

(*Launch multiple kernels*)
Get("AceFEM`");
kernels = ParallelEvaluate(
Get("AceFEM`Remote`");
$KernelID)

(*Solve FEM problem*)
Output=ParallelTable(
   
   {output1,output2}
   ,{problem,ListOfFEMProblems});

For problems that require the MATLAB script, I tried to modify the code to load MATLink and the MATLAB script in all cores in two different ways.

Focus 1:

(*Launch multiple kernels*)
Get("AceFEM`");
kernels = ParallelEvaluate(
Get("AceFEM`Remote`");
$KernelID)

(*Load MATLink*)
Needs("MATLink`");
OpenMATLAB();
MatlabFunction = MFunction("MatlabFunctionName", "OutputArguments" -> 2);

(*Solve FEM problem*)
Output=ParallelTable(
   {MATLABout1,MATLABout2}=MatlabFunction(Param1,Param2,Param3,Param4);
   
   {output1,output2}
   ,{problem,ListOfFEMProblems});

With the following error:

(kernel 4) Set::shape :  Lists {MATLABout1,MATLABout2} and     MATLink`MFunction(MatlabFunctionName,OutputArguments->2)(Param1,Param2,Param3,Param4) are not the same shape.

Focus 2:

(*Launch multiple kernels*)
Get("AceFEM`");
kernels = ParallelEvaluate(
Get("AceFEM`Remote`");
$KernelID)

(*Load MATLink*)
ParallelNeeds("MATLink`");
ParallelEvaluate(
   OpenMATLAB();
   MatlabFunction = MFunction("MatlabFunctionName", "OutputArguments" -> 2);
   );

(*Solve FEM problem*)
Output=ParallelTable(
   {MATLABout1,MATLABout2}=MatlabFunction(Param1,Param2,Param3,Param4);
   
   {output1,output2}
   ,{problem,ListOfFEMProblems});

With the mistakes:

(kernel 1) Global`OpenMATLAB::shdw :  Symbol OpenMATLAB appears in multiple contexts {Global`,MATLink`}; definitions in context Global` may shadow or be shadowed by other definitions.
(kernel 1) Global`MFunction::shdw :  Symbol MFunction appears in multiple contexts {Global`,MATLink`}; definitions in context Global` may shadow or be shadowed by other definitions.

Appears for each of the 4 kernals. As well as what happened previously Set::shape error.

(For clarity, calling the MATLAB function works perfectly for non-parallel problems)

What is the correct approach to MATLink parallelization?

Useful functions by searching (super button)

When I press start and start typing the name of an application to search for it, is there a way to allow this to reveal answers to questions like math, definitions and unit conversion similar to the search for the Windows 10 start button?

javascript: how to import a JS file and cancel one of its functions in Typecript

Recently I started with Angular and Typecript. My question is similar to: Overriding class methods based on JavaScript functions in TypeScript

There is clear OP about the implementation and the method to be overridden. In my case, I am using a third-party library that has a file dropdown.js that relates to DropDown component that I am using in my angular project. Your js code has an error. I have informed him several times, but they seem to care nothing. Is there any way to override the method in my typing and correct that error? Here it is his implementation of:

dropdown.js

"use strict";
var __extends = (this && this.__extends) || (function() {
    var extendStatics = Object.setPrototypeOf ||
        ({ __proto__: () }
            instanceof Array && function(d, b) { d.__proto__ = b; }) ||
        function(d, b) {
            for (var p in b)
                if (b.hasOwnProperty(p)) d(p) = b(p);
        };
    return function(d, b) {
        extendStatics(d, b);

        function __() { this.constructor = d; }
        d.prototype = b === null ? Object.create(b) : (__.prototype = b.prototype, new __());
    };
})();
Object.defineProperty(exports, "__esModule", { value: true });
var dropdown_base_1 = require("./dropdown.base");
require("./dropdown.scss");
var DropDownComponent = /** @class */ (function(_super) {
    __extends(DropDownComponent, _super);

    function DropDownComponent(_control, popup, _renderer, element) {
        ...
    }

    // ******* THIS METHOD I WANT TO OVERRIDE*********
    DropDownComponent.prototype.writeValue = function(value) {
        if (this.options) {
            this.selectOptionByValue(value);
        }
    };

    return DropDownComponent;
}(dropdown_base_1.DropDownBase));
exports.DropDownComponent = DropDownComponent;

And my typed file is quite simple:

mymenu.component.ts

import { Component, OnInit } from '@angular/core';
import {DropDowncomponent} from './../lirbray/dropdown'

@Component({
    selector: 'app-mymenu',
    templateUrl: './mymenu.component.html'
})
export class MyMenuComponent implements OnInit {
    constructor() {}

    ngOnInit(): void {
      this.translateModes();
    }
}

How do I import your javascript and cancel this method?

    DropDownComponent.prototype.writeValue = function(value) {
        if (this.options) {
            this.selectOptionByValue(value);
        }
    };

Is this feasible or am I asking a stupid question? Correct me because I'm from C, C ++ and I'm new to JavaScript. I have been blocked for this error for 1 month. Please help.