Polynomials and delta-constants.
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},
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"fields_dict": {
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"Basic": [
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---
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title: "2024-04-18"
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---
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- [x] Anki Flashcards
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- [x] KoL
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- [ ] Sheet Music (10 min.)
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- [ ] Go (1 Life & Death Problem)
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- [ ] Korean (Read 1 Story)
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- [ ] Interview Prep (1 Practice Problem)
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- [ ] Log Work Hours (Max 3 hours)
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@ -0,0 +1,15 @@
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---
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title: "2024-04-19"
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---
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- [x] Anki Flashcards
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- [x] KoL
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- [ ] Sheet Music (10 min.)
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- [ ] Go (1 Life & Death Problem)
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- [ ] Korean (Read 1 Story)
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- [x] Interview Prep (1 Practice Problem)
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- [x] Log Work Hours (Max 3 hours)
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* Hide-and-seek application
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* Revisions to settings. Primarily flattening the structure.
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* Read 2.3 "Polynomial Fitting" of "Discrete Mathematics: An Open Introduction".
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---
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title: "2024-04-18"
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---
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- [x] Anki Flashcards
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- [x] KoL
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- [ ] Sheet Music (10 min.)
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- [ ] Go (1 Life & Death Problem)
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- [ ] Korean (Read 1 Story)
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- [x] Interview Prep (1 Practice Problem)
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- [x] Log Work Hours (Max 3 hours)
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* Discussed first chapter of "Grokking the System Design Interview" as well as chapter 6 of "Designing Data-Intensive Applications" with Mike.
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* Applied to several more job openings.
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* Revised presence tracking and host updates in hide-and-seek application.
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---
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title: Polynomials
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TARGET DECK: Obsidian::STEM
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FILE TAGS: algebra::polynomial
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tags:
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- algebra
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- polynomial
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---
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## Overview
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Given nonnegative integer $d$, a **polynomial in $n$ of degree $d$** is a function $p(n)$ of the form $$p(n) = \sum_{i=0}^d a_i n^i$$
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The coefficients of $p(n)$ are $a_0, a_1, \ldots, a_d$. Furthermore, $a_d \neq 0$.
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%%ANKI
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Basic
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Using sigma notation, a polynomial in $n$ of degree $d$ is a function of what form?
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Back: $p(n) = \sum_{i=0}^d a_in^i$ where $a_d \neq 0$.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1713580808758-->
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END%%
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%%ANKI
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Basic
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What four algebraic operations are permitted in a polynomial?
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Back: Addition, subtraction, multiplication, and exponentiation.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1713580808763-->
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END%%
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%%ANKI
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Basic
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What is $d$ in "a polynomial in $n$ of degree $d$"?
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Back: $d$ is a nonnegative integer.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1713580808766-->
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END%%
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%%ANKI
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Basic
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What is $n$ in "a polynomial in $n$ of degree $d$"?
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Back: The polynomial's indeterminate.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1713580808769-->
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END%%
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%%ANKI
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Basic
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Which coefficient is special in a polynomial and why?
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Back: That attached to the monomial with highest degree because it cannot be zero.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1713580808772-->
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END%%
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%%ANKI
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Basic
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What are the coefficients of a polynomial?
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Back: The constants of the monomials found in the polynomial.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1713580808776-->
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END%%
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%%ANKI
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Basic
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What name is given to a degree-0 polynomial?
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Back: A constant.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1708974221749-->
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END%%
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%%ANKI
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Basic
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What name is given to a degree-1 polynomial?
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Back: A monomial.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708974221752-->
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END%%
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%%ANKI
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Basic
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What name is given to a degree-2 polynomial?
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Back: A binomial.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708974221755-->
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END%%
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%%ANKI
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Basic
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What is a binomial?
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Back: A polynomial containing two terms.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708368078759-->
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END%%
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%%ANKI
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Basic
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What is polynomial fitting?
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Back: The solving of a linear system to find the coefficients of a polynomial.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1713580109018-->
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END%%
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%%ANKI
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Basic
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Polynomial fitting is a strategy for discovering what part of a polynomial?
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Back: The coefficients.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1713580808780-->
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END%%
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%%ANKI
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Basic
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What is the degree of a polynomial?
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Back: The highest degree of the monomials with non-zero coefficients.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1713580109082-->
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END%%
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%%ANKI
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Basic
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How many data points are required to fit a polynomial?
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Back: $k + 1$ where $k$ is the degree of the polynomial.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1713580109089-->
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||||
END%%
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## Bibliography
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* Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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* Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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@ -150,6 +150,14 @@ Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n
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<!--ID: 1709666305449-->
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END%%
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%%ANKI
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Basic
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Why is a sequence of partial sums named the way it is?
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Back: Each term is found by adding a finite number of infinite terms.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1713580109297-->
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END%%
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## Bibliography
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* Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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@ -0,0 +1,201 @@
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---
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title: ∆ᵏ-constant Sequence
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TARGET DECK: Obsidian::STEM
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FILE TAGS: algebra::sequence algebra::polynomial
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tags:
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- algebra
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- sequence
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---
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## Overview
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Let $(a_n)$ be a sequence. We define the **$k$th differences of $(a_n)$** recursively:
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* The $0$th differences of $(a_n)$ is $(a_n)$.
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* The $k$th differences of $(a_n)$ is the sequence given by subtracting consecutive terms of the $(k-1)st$ differences of $(a_n)$.
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A sequence is said to be **$\Delta^k$-constant** if the $k$th differences are constant.
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> The closed formula for a sequence will be a degree $k$ polynomial if and only if the sequence is $\Delta^k$-constant.
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This is the discrete analogue to (continuous) derivatives of polynomials.
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%%ANKI
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Basic
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What are the $0$th differences of $(a_n)_{n \geq 0}$?
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Back: $(a_n)_{n \geq 0}$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1713580109096-->
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END%%
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%%ANKI
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Basic
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What are the $1$st differences of $(a_n)_{n \geq 0}$?
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Back: $(b_n)_{n \geq 1}$ where $b_n = a_n - a_{n - 1}$.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1713580109118-->
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END%%
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%%ANKI
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Basic
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How do we refer to the $1$st differences of $(a_n)$ more naturally?
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Back: As "the differences of $(a_n)$."
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1713580109125-->
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END%%
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%%ANKI
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Basic
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What is the base case of the recursive definition of the $k$th differences of $(a_n)$?
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Back: $k = 0$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1713580109133-->
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END%%
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%%ANKI
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Basic
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What is the recurrence of the recursive definition of the $(k + 1)$st differences of $(a_n)$?
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Back: The $(k + 1)$st differences is the differences of the $k$th differences.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1713580109153-->
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END%%
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%%ANKI
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Basic
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What does it mean for $(a_n)$ to be $\Delta^k$-constant?
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Back: The $k$th differences of $(a_n)$ is constant.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1713580109175-->
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END%%
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%%ANKI
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Basic
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What can be said about the closed formula of a $\Delta^k$-constant sequence?
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Back: It is a polynomial with degree $k$.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1713580109182-->
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END%%
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%%ANKI
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Cloze
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We say a sequence is {$\Delta^k$-constant} when the {$k$th differences is constant}.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1713580109189-->
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END%%
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%%ANKI
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Basic
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$\Delta^k$-constant sequences are a discrete analogue to what calculus concept?
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Back: Derivatives.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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Tags: calculus
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<!--ID: 1713580109197-->
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END%%
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%%ANKI
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Basic
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What kind of mathematical expression do $\Delta^k$-constant sequences relate to?
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Back: Polynomials.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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Tags: calculus
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<!--ID: 1713580109203-->
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END%%
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%%ANKI
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Basic
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How can we prove every quadratic sequence, say $(a_n)$, has arithmetic differences?
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Back: By showing $a_{n+1} - a_n$ is linear.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1713580109225-->
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END%%
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%%ANKI
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Basic
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How can we prove every cubic sequence, say $(a_n)$, has quadratic differences?
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Back: By showing $a_{n+1} - a_n$ is quadratic.
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||||
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
|
||||
<!--ID: 1713580109232-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Cloze
|
||||
The closed formula for a sequence will be a {degree $k$ polynomial} if and only if the $k$th differences {is constant}.
|
||||
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
|
||||
<!--ID: 1713580109237-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
How are arithmetic sequences defined in terms of $\Delta^k$ polynomials?
|
||||
Back: A sequence is arithmetic if and only if it is $\Delta^1$-constant.
|
||||
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
|
||||
<!--ID: 1713580109244-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
How are geometric sequences defined in terms of $\Delta^k$ polynomials?
|
||||
Back: N/A
|
||||
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
|
||||
<!--ID: 1713580109250-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
*Why* can't a geometric sequence be $\Delta^k$-constant for some $k \geq 0$?
|
||||
Back: Because the closed formula of a geometric sequence is not a polynomial.
|
||||
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
|
||||
<!--ID: 1713580109255-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
Suppose $(a_n)$ is a $\Delta^2$-constant sequence. What general form describes its closed formula?
|
||||
Back: $an^2 + bn + c$
|
||||
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
|
||||
<!--ID: 1713580109261-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
Suppose $(a_n)$ is a $\Delta^3$-constant sequence. What general form describes its closed formula?
|
||||
Back: $an^3 + bn^2 + cn + d$
|
||||
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
|
||||
<!--ID: 1713580109265-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
Suppose $(a_n)$ is a $\Delta^k$-constant sequence. Where is $k$ repeated in $(a_n)$'s closed formula?
|
||||
Back: At the largest degree of the polynomial.
|
||||
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
|
||||
<!--ID: 1713580109270-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
What are the terms in a $\Delta^0$-constant sequences?
|
||||
Back: $c, c, c, \ldots$ for some constant $c$.
|
||||
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
|
||||
<!--ID: 1713580109274-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
How do $\Delta^k$-constant sequences relate to polynomial fitting?
|
||||
Back: The closed formula of a $\Delta^k$-constant sequence is a polynomial we can fit.
|
||||
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
|
||||
<!--ID: 1713580109280-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
How many data points of a $\Delta^k$-constant sequence are needed to polynomial fit its closed formula?
|
||||
Back: $k + 1$
|
||||
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
|
||||
<!--ID: 1713580109285-->
|
||||
END%%
|
||||
|
||||
## Bibliography
|
||||
|
||||
* Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
|
||||
|
|
@ -254,6 +254,14 @@ Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://e
|
|||
<!--ID: 1709419325981-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
The triangular numbers is the sequence of partial sums of what sequence?
|
||||
Back: $a_n = n$
|
||||
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
|
||||
<!--ID: 1713580109292-->
|
||||
END%%
|
||||
|
||||
## Bibliography
|
||||
|
||||
* Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
|
||||
|
|
|
|||
|
|
@ -525,46 +525,6 @@ Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n
|
|||
<!--ID: 1708368078753-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
What is a degree-0 polynomial?
|
||||
Back: A constant.
|
||||
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
|
||||
<!--ID: 1708974221746-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
What name is given to a degree-0 polynomial?
|
||||
Back: A constant.
|
||||
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
|
||||
<!--ID: 1708974221749-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
What name is given to a degree-1 polynomial?
|
||||
Back: A monomial.
|
||||
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
|
||||
<!--ID: 1708974221752-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
What name is given to a degree-2 polynomial?
|
||||
Back: A binomial.
|
||||
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
|
||||
<!--ID: 1708974221755-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
What is a binomial?
|
||||
Back: A polynomial containing two terms.
|
||||
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
|
||||
<!--ID: 1708368078759-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
Why are binomial coefficients named the way they are?
|
||||
|
|
|
|||
Loading…
Reference in New Issue