Tagged Questions

A lattice in $\mathbb R^n$ is a discrete subgroup of $\mathbb R^n$ or, equivalently, it is a subgroup of $\mathbb R^n$ generated by linearly independent vectors. Lattices have applications in geometric number theory, e.g. via Minkowski's theorem.

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Naive Grouping for Factorization

I have a naive grouping method for factorization. I am curious as to its novelty and aspects of the code below that will increase its efficiency. The method is best described with an example: For n ...
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Is $\mathbb{Z}[\sqrt{2}]$ a lattice?

Is $\mathbb{Z}[\sqrt{2}]=\{a+b\sqrt{2}\mid a,b\in\mathbb{Z}\}$ a discrete subgroup of $\mathbb{R}$? How to prove that?
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Solve the indeterminate equation: $ad-bc=p$ for a prime integer $p$

How to solve the indeterminate equation: $ad-bc=p$ for a prime integer $p$? The origin of this problem is the following question: Show that rank-2 free $\mathbb Z$ module $\mathbb Z^2$ has $p+1$ ...
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Sacle the distance of lattice points

I know that for a hexagonal lattice generated by (0,1) and ($\sqrt{3}/2$,1/2) (i.e., when the distance between lattice points is 1), the number of lattice points in a circle of given radius $r$ can be ...
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Determining if $\mathbb{Z}[a]$ is a discrete subring of $\mathbb{C}$.

Let $a \in \mathbb{C}$ and consider the ring $\mathbb{Z}[a]$. Is there some nice criterion which will tell me whether $\mathbb{Z}[a]$ is discrete in the sense that there is some $\delta >0$ such ...
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Possible areas within an integer grid

Given a 1x1 grid with 4 lattice points $[(0,0),(0,1),(1,0),(1,1)]$ (equivalent to a $2 \times 2$ grid of vertices), there are 2 shapes and areas that can be formed: a triangle and a square. There are ...
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lattice walks with primes and composites

In the regular square lattice create a walk moving according the value of a counter. Consider two types of walks: In the first walk advance forward one unit if the counter is a composite number and ...
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Volume of the lattice generated by an ideal

Let $F$ be a totally real number field, $\mathfrak a \subset F$ a fractional ideal. Consider a lattice in $\mathbb R^n$ consisting of vectors $(\sigma_1(v),..\sigma_n(v))$, where $\sigma_1,..\sigma_n$ ...
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Which planar angles on an integer lattice are possible?

As shown in this question, you can construct an angle $A$ on 3 integer points on a plane only if $\tan A$ is rational. A natural generalization is to ask which values can planar angles based on 3 ...
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Boundedness condition of Minkowski's Theorem

Statement: "Let L be a lattice in $R^n$ and $S\subset R^n$ be a convex, bounded set symmetric about the origin. If $Volume(S) > 2^ndet(L)$, then S contains a nonzero lattice vector. Moreover, if ...
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What is the significance of $SL(2, \mathbb{R} / SL(2, \mathbb{Z}))$ in studying lattices in geometry of numbers?

I was listening to a talk about lattices and the geometry of numbers and at one point they jumped from discussing a 2d lattice into discussing $SL(2, \mathbb{R})\ /\ SL(2, \mathbb{Z}))$ and it was not ...
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Prove that $a_{i,1}x_1 +a_{i,2}x_2 +···+a_{i,n}x_n ≤c_i, 1≤i≤n$ are all satisfied by a nonzero $n-tuple$ of integers.

My setting is that $c_1, · · · c_n$ are positive real numbers, and $A = [a_{i,j} ]$ is an $n × n$ non-singular matrix. Assume that $c_1 · · · c_n > | det(A)|.$ I want to prove that the n-linear ...
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Is it true that for a lattice $L$, $\mathbb{R}L = \mathbb{R}^{n}$?

I have as a definition A lattice $L \subseteq \mathbb{R}^{n}$ is a subgroup that is free of rank $n$ such that $\mathbb{R}L = \mathbb{R}^{n}$. I don't know if I am misinterpreting the statement, ...
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Find a function so whenever it is near a lattice point $\lim_{x \rightarrow [x_0]}f(x)=[y_0]$

A function $f:\mathbb{R} \rightarrow \mathbb{R}$ is an "easy estimator" if any point on $f$, $(x_0,y_0)$, is near a lattice point $([x_0],[y_0])$ then $\lim_{x \rightarrow [x_0]}f(x)=[y_0]$. In ...
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When does a real inverrtible matrix send a lattice in $\mathbb{R}^2$ to itself?

There is a question in Artin asking when does an inverrtible 2x2 real matrix send a lattice to itself. The issue I have is with this question not being specific as to what kind of answer is expected....
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Finding an integral basis for a lattice defined in terms of an equation modulo p

Let p be a prime number, u relatively prime to p, and $\Lambda := \lbrace (a, b) \in \mathbb{Z}^2 : b \equiv au$ (mod p)$\rbrace$. How then can I find an integral basis $v_1, v_2$ for $\Lambda$? I ...
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How to generally describe all possible quasi-crystal structures in $\mathbb{R}^3$?

According to what I found on Wikipedia[1,2], you can represent any quasi-crystal structure in $\mathbb{R}^n$ by cutting a space $\mathbb{R}^N, N>n$ at an angle with the $\mathbb{R}^n$ space and ...
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For a non-square integer $d$ such that $d \equiv 1 \mod 4,$ we define the set $$\mathcal{O}_d := \left\{\frac{a + b\sqrt{d}}{2}:a,b \in \mathbb{Z}, a \equiv b \mod 2 \right\}.$$ Prove that $\... 1answer 45 views Maximal even sub-lattice in$\mathbb{Z}^n$(reference request) Lemma: The maximal even sub-lattice in$\mathbb{Z^n}$is $$\Big\{ (x_1,\cdots,x_n)\in \mathbb{Z}^n ~~\big|~~ \sum_{1 \le i \le n} x_i \in \mathbb{2Z} \Big\}$$ I found the above lemma in an ... 0answers 81 views Minimize multivariate (multivariable) polynomial over the integers I'd like to minimize the following polynomial in 6 variables$h_0,h_1,g_0,g_1,g_2,g_3$: $$g_3^2\cdot h_0^3\cdot h_1^3 - g_2\cdot g_3\cdot h_0^2\cdot h_1^4 + g_1\cdot g_3\cdot h_0\cdot h_1^5 - g_0\... 1answer 56 views How do we balance the chemical equation and give an integer solution? [closed]$$\require{mhchem}\ce{NaOH + H2SO4 -> Na2SO4 + H2O}$$I'm trying all my best to come up with one solution but cant. in giving the integer solution to this chemical equation is it compulsory? 1answer 34 views Why ins't \mathfrak{h} enough to parametrize complex elliptic curves? this a pretty idiot question and of course there is a mistake in my way of thinking. Let E be a elliptic curve, E (\mathbb{C}) \cong \mathbb{C} / \Lambda, where \Lambda = \langle \omega_1, \... 0answers 22 views Coexistance of certain vectors in a lattice I'm currently working on the SVP (Shortest Lattice Vector Problem) as a part of a paper that I'm writing. I've been trying to prove ( or disprove) the following without too much success : Question : ... 1answer 21 views Is it easy to find a lattice vector whose length is in a specific interval? Say L is a lattice of {\bf R}^n with rank m. Let \alpha, \beta \ (\alpha<\beta) be positive real numbers. Set A_{L}=\{{\bf x}\in L: \alpha\leq \|{\bf x}\|\leq \beta\}. It may be difficult ... 1answer 127 views Finding a basis for the integer lattice points in a subspace When writing this answer, one subgoal involved finding all integer solutions to the equations \sum_{i=1}^5x_i=\sum_{i=1}^5ix_i=0 in \Bbb Z^5. Since this is a linear system, the solutions form a ... 0answers 68 views Lower bounds on possible integer relations from the PSLQ algorithm For the equation:$$ \sum_{i=1}^na_ix_i=0 $$where all x_i are real numbers and all a_i are integers, the PSLQ algorithm can either find an integer relation or give lower bounds on the norm of ... 1answer 41 views Measure theoretic proof of |\Bbb{Z}^d/A\Bbb{Z}^d| = |\det(A)| Let A \in \Bbb{Z}^{d\times d} be an invertible matrix with entries in \Bbb{Z}. It is well-known (and can be proved using algebraic properties of matrices) that the index of the group A \Bbb{Z}^d ... 2answers 73 views Lattice of integers \mathbf{Z} in \mathbb{R^2} Lattice of integers \mathbf{Z} in \mathbb{R^2} The questions: Give an example of a nonempty subset of \mathbb{R^2} (noted M) which is closed under addition and for all m\in M we have -m\in ... 1answer 53 views Integer solutions to a two variable equation. For m, n \in \mathbb{Z}, show the only integer solutions to f(m,n) = \displaystyle \frac{3^m(2^n+1)-2^{m+n}}{2^{m+n}-3^{m+1}} are f(1, 2) = -7, f(0, 1) = -1, and f(0, 2) = 1. More ... 1answer 121 views Why is Minkowski's Theorem so powerful? Minkowksi's Convex Body Theorem is evidently pretty powerful, as it yields swift proofs of Fermat's Two Square and Lagrange's Four Square Theorems. Also, Minkowski's bound on class number and the ... 0answers 131 views Finding a basis for a particular integer lattice The following problem arose in the context of string theory. I hope someone here might provide some guidance or a solution... Our starting point is: i) an integer-lattice L\subset\mathbb R^n ... 1answer 98 views Simple question about lattices in \mathbb{C} Milne's Modular Functions and Modular Forms states, at the bottom of page 10: We can normalize our lattices so they are of the form$$\Lambda(\tau) := \mathbb{Z} \cdot 1 + \mathbb{Z} \cdot \tau$$... 1answer 106 views How to eliminate some edges of a lattice to get exactly k paths? We have an n by n lattice. We want to find a way to eliminate some edges, so that there are exactly k paths from (1,1) to (n,n) of length 2n-2. (this means our paths should be NE). I don't ... 1answer 130 views Lattice in \Bbb R^2 Let (a,b) be a lattice basis of a lattice L in \mathbb{R}^2. Prove that every other lattice basis has the form (a',b')=(a,b)P, where P is a 2\times 2 integer matrix with determinant 1 or ... 2answers 336 views Proving that T:(x_1,…,x_n) \rightarrow (\frac {x_1+x_2}{2},\frac {x_2+x_3}{2},…,\frac {x_n+x_1}{2}) leads to nonintegral components Start with n paiwise different integers x_1,x_2,...,x_n,(n>2) and repeat the following step: T:(x_1,...,x_n) \rightarrow (\frac {x_1+x_2}{2},\frac {x_2+x_3}{2},...,\frac {x_n+x_1}{2}) ... 1answer 90 views Number of Dyck paths from (0,0) to (2n,k_1) if allowed to go below the x axis What is the number of (general?) Dyck paths from (0,0) to (2n,k_1), where k_1\geq0, allowing the path to go below the x axis and touch the negative horizontal line at k_2\leq0 an arbitrary ... 1answer 107 views Lattice generated by vectors orthogonal to an integer vector Given a non-zero vector \boldsymbol{v} composed of integers, imagine the set of all non-zero integer vectors \boldsymbol{u}, such that \boldsymbol{u} \cdot \boldsymbol{v} = 0, i.e., the integer ... 1answer 42 views Number of solution pairs (i,j) such that i+jk \leq l I have show that the number of solutions \left(\, i,j\,\right) of non-negative integers to i + jk \leq l is$$ \left(\,\left\lfloor\, l \over k\,\right\rfloor +1\,\right) {2l + 2 - k\left\lfloor\, ... 0answers 78 views Line and distance to lattice points? Consider the lattice$\mathbb{Z}^{n}\subset\mathbb{R}^{n}$and a line (not necessarily through the origin). What conditions can be placed on the slope of the line that is necessary and sufficient so ... 0answers 100 views Lattices in$\mathbb C$as modules of the ring of integers in an imaginary quadratic field Let$K$be an imaginary quadratic number field and let$O_K\subset K$be the ring of algebraic integers in$K$. Let us call a lattice$\Lambda\subset\mathbb C$normalized if the tori$\mathbb C/\...
I was wondering how many self-avoiding rook walks there are on an $m×n$ grid. A self-avoiding rook walk is a path from the bottom left corner to the top right corner of the grid, composed only of ...