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# Products taken over semiprimes

I was inspired in the section B48 of Guy's book ([1]) for ask myself a question that arises when one consider the sequence for integers $$m\geq 1$$

$$\mathcal{S}_m:=\prod_{n=1}^m\frac{s_n+1}{s_n-1}\tag{1}$$ where $$s_n$$ denotes the sequence of semiprimes, thus $$s_1=4$$, $$s_2=6$$, $$s_3=9\ldots$$, that is A001358 in the OEIS (see the Wikipedia Semiprime).

Seems that it is obvious that exists an integer $$m_0$$ for which $$\forall m\geq m_0$$ the denominator of $$S_m$$ (written as irreducible fraction) is an even integer, while that its numerator is odd, thus seems that the following conjecture is "obvious".

Conjecture. There exists an integer $$m_0$$ such that $$\forall m\geq m_0$$ $$\prod_{n=1}^m\frac{s_n+1}{s_n-1}$$ is never an integer.

What I am saying is that semiprimes as $$9,33$$ or $$129$$ add in the denominator of the corresponding $$S_m$$ powers of two with a great exponent. But I don't know how to get a rigorous proof from this heuristic. I've tested the conjecture for the first few dozens of semiprimes.

Question. What is/Is there a proof for the existence of an integer $$m_0$$ such that $$\forall m\geq m_0$$ $$\prod_{n=1}^m\frac{s_n+1}{s_n-1}$$ is never an integer? Many thanks.

I hope that this question isn't obvious, and good for this MathOverflow. Thanks all.

## References:

[1] Richard K. Guy, Unsolved Problems in Number Theory, Unsolved problems in Intuitive Mathematics Volume I, Second Edition, Springer-Verlag (1994).

• It is not clear to me why the sequence $\{s_n\}$ is particularly interesting in formulating a conjecture like this, rather than some other infinite sequence of positive integers. Aug 3 '19 at 13:10
• (1/2) Yes, @StanleyYaoXiao I am asking just about a sequence, because if with few computational evidence I ask about more sequences then the post doesn't seem serious. I thought, but I did not do the calculations for the sequence of square-free integers (the same product, specialization for squarefree integers, taken for $n\geq 2$), and I tried with variants of some of the best known sequences of figurate numbers, see the table in the encyclopedia MathWorld Figurate Number. Invoking a formula due to Sitaramachandrarao (see the Wikipedia Euler's totient function), then combining with the Aug 4 '19 at 8:08
• (2/2) convergence criteria for infinite products I know that $\prod_{n=2}^\infty\frac{n\varphi(n)+1}{n\varphi(n)-1}$ will be convergent, but I think that has no a nice closed-form. Thus this is an invitation if you, or some of your colleagues in MO want to study more variants (the problems that have a good mathematical content are those showed in Guy's book). With my computational evidence and mathematical experience, I could not propose a more interesting product. Many thanks again, as was said this is an invitation if some user want try to explore these products or variants of problems Aug 4 '19 at 8:08


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# In relativity, if units of length contracts and time dilates then does unit of velocity or speed also change?

I'm just starting to learn special relativity, and I'm having trouble with the following concept:

In relativity, units of length and time of moving frame are related to that of stationary one through $$x’=\frac{x}{\gamma}\quad \quad \text{ and }\quad \quad t’=t\times \gamma$$ respectively, where $$\gamma$$ is Lorentz Factor.

Does this also mean that units of velocity or speed, i.e. length/time are related as $$v’=\frac{x’}{t’}=\frac{x}{t}\times\frac{1}{\gamma^2}=\frac{v}{\gamma^2}?$$

Note: By unit, I mean scale of axes in a respective coordinate system and I am not asking about addition or subtraction of velocities, I am enquiring about mutual “scale” difference between the quantity called velocity as measured in two different frames in uniform relative motion to each other.

Why scale of length contracts and not expands while that of time dilates, i.e. expands when the two are symmetrical for Lorentz transformations! Only if length expands with dilation of time can the “scale” of velocity or speed in general and speed of light in particular can remain truly invariant, I guess.

• This question is very similar to this one: Time Dilation + Length Contraction Scenario, though it's worded a little differently. Check out my answer there. Aug 22 '20 at 6:17
• What you write is not a Lorentz transformation. Aug 22 '20 at 7:25
• Thank you so much Philip for typesetting and upgrading the language of question. Surely I’ll learn and use MathJax in future.
– rim
Aug 22 '20 at 8:31

No. Velocities do transform in a non-intuitive way in special relativity, but not in the way you're describing. This is sadly a very common misunderstanding due to the fact that Relativity is usually first introduced using time-dilation and length contraction, without actually explaining under which conditions they are applicable. The best way to begin understanding the subject (and to also avoid all these "paradoxes") is to work with the Lorentz Transformations. In one dimension, if the frame $$S'$$ is moving with respect to $$S$$ with a velocity $$v$$, then

\begin{aligned} x' &= \gamma\left( x - vt \right)\\ t' &= \gamma \left( t - \frac{v}{c^2}x\right) \end{aligned}

provided that $$x=x'=0$$ when $$t=t'=0$$. Remember, though, that these are coordinates, not intervals. To find how intervals of length and intervals of time are related, we need to take differences of the coordinates, and since $$v$$ and $$\gamma$$ are constants, it's easy to show that the intervals satisfy similar equations:

\begin{aligned} \Delta x' &= \gamma\left( \Delta x - v \Delta t \right)\\ \Delta t' &= \gamma\left( \Delta t - \frac{v}{c^2}\Delta x\right) \end{aligned}

We can use the above equations to easily calculate how the velocities transform between the frames $$S$$ and $$S'$$. Remember that an observer in $$S$$ will calculate the velocity of an object to be $$u = \frac{\Delta x}{\Delta t},$$ and one in $$S'$$ will calculate it to be $$u' = \frac{\Delta x'}{\Delta t'}.$$

We can now divide $$\Delta x'$$ by $$\Delta t'$$ to show that:

$$u' = \frac{\Delta x'}{\Delta t'} = \frac{u - v}{1 - \frac{uv}{c^2}}.$$

### Why doesn't your argument work?

Length contraction and time dilation are special cases of the general formulae that I have given above. They hold when certain conditions are satisfied, and these conditions are more certainly not satisfied simultaneously. Which means dividing the equations is not going to give you anything sensible. In special relativity, it is best to think in terms of "events" which occur at spacetime points $$(t, x)$$ to avoid such false "paradoxes". My answer here, and the links at the end, should explain it in more detail.

• First of all, thanks Philip for the reply, but I am afraid I am still unable to grasp the concept. The point of confusion is not the numerical aspect but the unit aspect. Correct me if I am wrong, any dimensional value has two things, a number and a unit, right! Now in one coordinate system velocity may be v meters/second and in other it may be v’ meter’/second’ with v & v’ related as per relation you have given. My confusion is how meter’/second’ be related to meter/second. Is it as per the relation in my question since meter’=meter/𝛾 & second’=secondX 𝛾?
– rim
Aug 22 '20 at 7:56
• @rim I see, I didn't understand that from the question. But that's really not a problem at all! $\gamma$ has no units (it's dimensionless, just a pure number). So $x$ and $x'$ have the same units (just as $t$ and $t'$ have the same units). All physical quantities (including velocities) have the same units in both frames. Does that help? Aug 22 '20 at 7:59
• Again thanks for reply Philip but sorry for my inability to understand, If units remain same between different coordinate systems then why do we say that length has contracted or time has dilated in the first place? i.e. why one meter measured by an observer is different from one meter measured by another observer in relative motion with former! During transformation of coordinates three things can happen, either number of unit can change or measure of unit itself can change or both. I am unable to grasp second & third aspect of Lorentz transformations, I guess.
– rim
Aug 22 '20 at 8:27
• A metre continues to be a metre, since it's just a measure of length. Let's take an example: you are in $S'$ moving with respect to me in $S$ with a speed $v$. Now, we both have metre scales in our hands. I measure my scale to be 1 metre in my frame, but I will measure your scale to be contracted to $1/\gamma$ metres. Similarly, you measure your own scale to be 1 metre, but you will measure mine to be $1/\gamma$ metres. Does this help? The unit itself doesn't change, it is the numerical value that does. Aug 22 '20 at 8:30
• It seems I am equally bad at conveying as at understanding! Doesn’t that mean “scale” of velocity or speed also changes, i.e. forget the numerical part, since my scales of meter and second seems different to you, won’t my scale for a “unit of” velocity (meter/second) will seem different to you (and that too by 1/𝛾^2)?! I may be measuring speed of light as c m/s but for you I must be measuring speed of light as c/𝛾^2 m/s, don’t you think? Please read “measure of unit” as scale in my previous comment.
– rim
Aug 22 '20 at 8:55

Along with Philip’s answer, this video ( https://youtu.be/-NN_m2yKAAk ) brought clarity to the problem.

Basically what I understood from the two is that time dilation and length contraction are not similar or symmetrical things (as I was assuming in original question) but by nature, we are able to measure time only using one and length using another method respectively.

In fact, there are concepts of length dilation and duration contraction that are not generally discussed but should be.


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# Light of frequency

Question:

Light of frequency $7.21 \times 10^{14} \mathrm{~Hz}$ is incident on a metal surface. Electrons with a maximum speed of $6.0 \times 10^{5} \mathrm{~m} / \mathrm{s}$ are ejected from the surface. What is the threshold frequency for photoemission of electrons?

Solution:

Frequency of the incident photon, $v=488 \mathrm{~nm}=488 \times 10^{-9} \mathrm{~m}$

Maximum speed of the electrons, v = 6.0 × 105 m/s

Planck’s constant, h = 6.626 × 10−34 Js

Mass of an electron, m = 9.1 × 10−31 kg

For threshold frequency ν0, the relation for kinetic energy is written as:

$\frac{1}{2} m v^{2}=h\left(v-v_{0}\right)$

$v_{0}=v-\frac{m v^{2}}{2 h}$

$=7.21 \times 10^{14}-\frac{\left(9.1 \times 10^{-31}\right) \times\left(6 \times 10^{5}\right)^{2}}{2 \times\left(6.626 \times 10^{-34}\right)}$

$=7.21 \times 10^{14}-2.472 \times 10^{14}$

$=4.738 \times 10^{14} \mathrm{~Hz}$

Therefore, the threshold frequency for the photoemission of electrons is $4.738 \times 10^{14} \mathrm{~Hz}$.


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# Sylow 7-subgroups in a group of order 168

Another question from my qual studying that's been stumping me. I'm still a little lost on normalizers. The question is:

Let G be a group of order 168, and let P be a Sylow 7-subgroup of G. Show that either P is a normal subgroup of G or else the normalizer of P is a maximal subgroup of G.

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The number of Sylow 7-subgroups is congruent to 1 mod 7, so is either 1 (when the Sylow subgroup is normal) or 8 (other possibilities are easy to dismiss). G acts on the Sylow 7-subgroups by conjugation, and the normaliser N of P is the subgroup of G which sends P to itself. This action is transitive on the 8 Sylow 7-subgroups so N has order 168/orbit of P = 168/8=21. There are limited options now - so consider those in turn. –  Mark Bennet Aug 13 '12 at 20:10

## 2 Answers

Suppose $P$ is not normal. Then by Sylow's theorems, $G$ has $8$ $7$-Sylows, all of them conjugate to $P$. Thus the normalizer $N$ of $P$ has index $8$ in $G$, and hence order $21$. Suppose $N$ is not maximal, so there's a subgroup $H$ of $G$ strictly between $N$ and $G$. Then $H$ must have order $42$ or $84$. Sylow's theorems show that $H$ has a normal $7$-Sylow, which must be $P$. But then $P$ is normal in $H$, contradicting the fact that $H$ properly contains the normalizer $N$ of $P$.

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A way to generalize is this: Let $G$ be a finite group and $P$ be a Sylow $p$-subgroup of $G$. Let $N= N_{G}(P).$ Then if $1 < [G:N] < (p+1)^{2},$ the subgroup $N$ must be maximal. For suppose that $M$ is a proper subgroup of $G$ which strictly contains $N.$ Then we have $[M:N] \equiv 1$ (mod $p$) by Sylow's theorem, since $N = N_{M}(P)$ and $P$ is still a Sylow $p$-subgroup of $M.$ Also, since $[G:N] \equiv 1$ (mod $p$) and $[M:N] \equiv 1$ (mod $p$), we must have $[G:M] \equiv 1$ (mod $p$). Since $[G:M] >1$ and $[M:N] >1$ by assumption, we have $[G:N] = [G:M][M:N] \geq (p+1)^{2}.$

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# The number of real roots of the equation $2^{2x^{2}-7x+5}=1$ is

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The number of real roots of the equation $$2^{2x^{2}-7x+5}=1$$ is
A) 0
B) 1
C) 2
D) 3
E) 4

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If $$2^{something} = 1$$, then something = 0.

$$2x^2-7x+5 = 0$$
$$2x^2-2x-5x+5 = 0$$
$$2x(x-1)-5(x-1) = 0$$
x = 1 or x = 5/2 = 2.5

2 real roots.

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# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

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Nagumo type existence results for second-order nonlinear dynamic BVPS. (English) Zbl 1072.34015
A time scale ${\Bbb T}$ is a nonempty closed subset of ${\Bbb R}$. This paper studies the second-order nonlinear dynamic equation $y^{\Delta\Delta}(t)=f(t,y^\sigma(t),y^\Delta(t))$ on time scales with certain functional boundary conditions. The authors introduce a Nagumo-type condition (restricting the growth of $f$ in $y^\Delta$) and establish the existence of at least one solution of the given boundary value problem lying between the associated lower and upper solutions. An example is presented illustrating the main result.

##### MSC:
 34B15 Nonlinear boundary value problems for ODE 39A12 Discrete version of topics in analysis
##### Keywords:
time scale; Nagumo condition; lower and upper solutions
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##### References:
 [1] A. Adje. Sur et Sous-Solutions dans les Equations Differentielles Discontinues avec des Conditions aux Limites non Linéaires, Ph. D., Université Catholique de Louvain-la-Neuve, 1987. [2] Bohner, M.; Peterson, A.: Dynamic equations on time scales, an introduction with applications. (2001) · Zbl 0978.39001 [3] Cabada, A.; Espinar, V. Otero --; Pouso, R. L.: Existence and approximation of solutions for discontinuous first order difference problems with nonlinear functional boundary conditions in the presence of lower and upper solutions. Comput. math. Appl. 39, 21-33 (2000) · Zbl 0972.39002 [4] A. Cabada, D.R. Vivero, Expression of the Lebesgue \Delta  -- integral on time scales as a usual Lebesgue integral, Application to the calculus of \Delta  -- antiderivatives, preprint. · Zbl 1092.39017 [5] Eloe, P. W.: A boundary value problem for a system of difference equations. Nonlinear anal. 7, 813-820 (1983) · Zbl 0518.39002 [6] D. Franco, D. O’Regan, Existence of solutions to second order problems with nonlinear boundary conditions, Dynamical Systems and Differential Equations (Wilmington, NC, 2002); Discrete Contininuous Dyn. Syst. Suppl. (2003) 273 -- 280. [7] Habets, P.; Pouso, R. L.: Examples on the nonexistence of a solution in the presence of upper and lower solutions. Anziam J. 44, 591-594 (2003) · Zbl 1048.34036 [8] Hilger, S.: Analysis on measure chains --- a unified approach to continuous and discrete calculus. Results math. 18, 18-56 (1990) · Zbl 0722.39001 [9] Kaymakçalan, B.; Laksmikantham, V.; Sivasundaram, S.: Dynamical systems on measure chains. (1996) [10] Ladde, G. S.; Lakshmikantham, V.; Vatsala, A. S.: Monotone iterative techniques for nonlinear differential equations. (1985) · Zbl 0658.35003 [11] Nagumo, M.: Ueber die differentialgleichung y″=f(x,y,$y^{\prime}$). Proc. phys.-math. Soc. Japan 19, No. 3, 861-866 (1935) · Zbl 63.1021.04 [12] Nagumo, M.: On principally linear elliptic differential equations of the second order. Osaka math. J. 6, 207-229 (1954) · Zbl 0057.08201 [13] Picard, E.: Sur lapplication des methodes d’approximations successives a l’étude de certaines équations différentielles ordinaires. J. de math. 9, 217-271 (1893) · Zbl 25.0507.02 [14] Dragoni, G. Scorza: Il problema dei valori ai limiti studiato in grande per gli integrali di una equazione differenzile del secondo ordine. Giornale mat. (Battaglini) 69, 77-112 (1931) · Zbl 0002.25702 [15] Zhuang, W.; Chen, Y.; Cheng, S. S.: Monotone methods for a discrete boundary problem. Comput. math. Appl. 32, 41-49 (1996) · Zbl 0872.39005


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Distance-Enhanced Graph Neural Network for Link Prediction
Yingce Xia · Yingce Xia
Link prediction, which is to predict the existence of a link/edge between two vertices in a graph, is a classical problem in machine learning. Intuitively, if it takes a long distance to walk from $u$ to $v$ along the existing edges, there should not be a link between them, and vice versa. This motivates us to explicitly combine the distance information with graph neural networks (GNNs) to improve link prediction. Calculating the distances between any two vertices (e.g., shortest path, expectation of random walk) in training is time consuming. To overcome this difficulty, we propose an anchor-based distance: First, we randomly select $K$ anchor vertices from the graph and then calculate the shortest distances of all vertices in the graph to them. The distance between vertices $u$ and $v$ is estimated as the average of their distances to the $K$ anchor vertices. After that, we feed the distance into the GNN module. Our method brings significant improvement for link prediction with few additional parameters. We achieved state-of-the-art result on the drug-drug-interaction (i.e., DDI) and protein-protein-association (i.e., PPA) tasks of OGB~\cite{hu2020ogb}. Our code is available at \url{https://github.com/lbn187/DLGNN}.


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# Does the GLOA have any advantage over the Solovay-Kitaev algorithm?

The Solvay Kitaev algorithm was discovered long before the Group Leaders Optimization algorithm and it has some nice theoretical properties. As far as I understand, both have exactly the same goals: given a finite dimensional unitary operator, they decompose the operator into basic quantum gates. I couldn't find any theoretical results about the time complexity or convergence time or error bounds for the GLOA as such. Does the latter (GLOA) have any practical advantage over the former at all, in terms of convergence time or anything?

P.S: For a detailed description of the GLOA, see: Understanding the Group Leaders Optimization Algorithm

• I don’t know anything about GLOA, but nowadays there are better algorithms than Solovay-Kitaev – DaftWullie Oct 22 '18 at 5:46
• As a side question, do you have any link to resources about these algorithms? Or at least their name? @DaftWullie – Adrien Suau Oct 22 '18 at 6:53
• @Nelimee Sure, I just had to be on a decent computer to find the link I had. Try this one: arxiv.org/abs/1212.6253 – DaftWullie Oct 22 '18 at 8:50
• Thanks! But even if the algorithm is better in terms of complexity, it does not replace the SK algorithm. I though you were speaking of an algorithm as general as SK. – Adrien Suau Oct 22 '18 at 9:32
• @Nelimee the Ross-Selinger algorithm is not just better in terms of complexity than the Solvay-Kitaev algorithm, it has been implemented in practice and is very fast (were talking seconds to achieve errors in the $10^{-3}$ range and is the standard tool for doing approximate circuit synthesis on Clifford+T gate sets. In what way does it not replace the SK algorithm? – Condo Jul 7 '20 at 19:01

On the other hand the SK algorithm is an exact or deterministic algorithm with polynomial complexity in $$O(\log(1/\epsilon))$$ where $$\epsilon$$ is the desired accuracy of the quantum gate. The SK algorithm (unlike heuristic algorithms) takes advantage of the fact that quantum gates are elements of the unitary group $$U(d)$$, which is a Lie group and therefore a smooth manifold in which one can make conclusions about the geometry and distance between points.
GLOA's may have advantages when engineering quantum circuits as it may be possible to incorporate other desired aspects of circuit design into the optimization. However, they will not be as efficient nor optimal as the SK algorithm (or the more powerful number theoretic approximation algorithms for $$U(d)$$).


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Research Article| Volume 13, P82-89, November 2014

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# Verifying likelihoods for low template DNA profiles using multiple replicates

Open AccessPublished:July 09, 2014

### Highlights

• The behaviour of multi-replicate LRs with respect to the inverse match probability is proposed as a method to validate forensic LR software.
• We perform lab-based and simulated experiments of one-, two- and three-contributor CSPs, as well as investigating a real-world CSP.
• LRs rise towards the IMP with additional replicates, while never exceeding it. Additionally, the LR from multiple low-template replicates can exceed that from a single good-quality sample.
• We validate likeLTD by demonstrating that it adheres to the expected behaviours.

## Abstract

To date there is no generally accepted method to test the validity of algorithms used to compute likelihood ratios (LR) evaluating forensic DNA profiles from low-template and/or degraded samples. An upper bound on the LR is provided by the inverse of the match probability, which is the usual measure of weight of evidence for standard DNA profiles not subject to the stochastic effects that are the hallmark of low-template profiles. However, even for low-template profiles the LR in favour of a true prosecution hypothesis should approach this bound as the number of profiling replicates increases, provided that the queried contributor is the major contributor. Moreover, for sufficiently many replicates the standard LR for mixtures is often surpassed by the low-template LR. It follows that multiple LTDNA replicates can provide stronger evidence for a contributor to a mixture than a standard analysis of a good-quality profile. Here, we examine the performance of the likeLTD software for up to eight replicate profiling runs. We consider simulated and laboratory-generated replicates as well as resampling replicates from a real crime case. We show that LRs generated by likeLTD usually do exceed the mixture LR given sufficient replicates, are bounded above by the inverse match probability and do approach this bound closely when this is expected. We also show good performance of likeLTD even when a large majority of alleles are designated as uncertain, and suggest that there can be advantages to using different profiling sensitivities for different replicates. Overall, our results support both the validity of the underlying mathematical model and its correct implementation in the likeLTD software.

## 1. Introduction

In forensic DNA profiling, a likelihood ratio (LR) is calculated to measure the support provided by DNA evidence (E) for a proposition Hp favouring the prosecution case, relative to its support for Hd representing the defence case. The LR can be written as
$LR=Pr(E|Hp)Pr(E|Hd).$
(1)

Each of Hp and Hd specifies a number of unprofiled contributors and a list of contributors whose DNA profiles are known (included in E). Typically Hp includes a profiled, queried contributor that we designate Q, who is replaced under Hd by an unprofiled individual X. Q may be an alleged offender, or a victim, while X is an alternative, usually unknown, possible source of the DNA. It usually suffices to limit attention to Hp and Hd that differ only in replacing Q with X, otherwise the LR is difficult to interpret as a measure of the weight of evidence for Q to be a contributor of DNA.
In addition to reference profile(s), of Q and possibly other known contributors, the DNA evidence consists of one or more profiling runs performed on a DNA sample recovered from a crime scene, or from an item thought to have been present when the crime occurred. Each profiling run generates graphical results in an electropherogram (epg), which we assume has been interpreted by a forensic scientist who decides a list of alleles observed at each locus, and also a list of potential alleles about which there is substantial uncertainty, perhaps due to possible stutter. Alleles not on either list are regarded as unobserved in that run.
In low-template DNA (or LTDNA) profiling, each epg can be affected by stochastic effects such as dropin, dropout and stutter [
• Steele C.D.
• Balding D.J.
Statistical evaluation of forensic DNA profile evidence.
]. To help assess stochastic effects, it is common to perform multiple profiling runs, possibly varying the laboratory conditions but these are nevertheless referred to as replicates. Joint likelihoods for multiple replicates are obtained by assuming that the replicates are independent conditional on the genotypes of all contributors and parameters ϕ such as the amounts and degradation levels of DNA from each contributor [
• Curran J.M.
• Gill P.
• Bill M.R.
Interpretation of repeat measurement DNA evidence allowing for multiple contributors and population substructure.
]. We can write
$Pr(E|H)=∑jPr(Gj)∏iPr(Ri|Gj,ϕ),$
(2)

where Ri is the set of allele designations in the ith replicate run of the crime scene profile (CSP), $Gj$ denotes the jth set of contributor genotypes, and the summation is over all possible sets of contributor genotypes under H. Pr$(Gj)$ is computed under a standard population genetics model [
• Steele C.D.
• Balding D.J.
Statistical evaluation of forensic DNA profile evidence.
]. The unknown parameters ϕ can be replaced with estimates, or eliminated by maximisation or integration with respect to a prior distribution.
Currently, there are only limited possibilities to check the validity of an algorithm for evaluating an LTDNA LR (henceforth ltLR). One approach is to evaluate the ltLR when Q is repeatedly replaced by a random profile [
• Gill P.
• Haned H.
A new methodological framework to interpret complex DNA profiles using likelihood ratios.
]. In that case Hp is false and we expect the majority of computed ltLRs to be small. Here, we propose to investigate a performance indicator for ltLR algorithms when Hp is true. Under Hd, it may occur that $GX=GQ$, where $GX$ and $GQ$ denote the genotypes of X and Q. This occurs with probability πQ, the match probability for Q. Since Pr$(E|Hd,GX=GQ)=Pr(E|Hp)$, it follows that [
• Cowell R.G.
• Graversen T.
• Lauritzen S.
• Mortera J.
Analysis of DNA mixtures with artefacts.
]
$ltLR=Pr(E|Hp)Pr(E|Hd,GX=GQ)πQ+Pr(E|Hd,GX≠GQ)(1−πQ)≤1πQ.$
(3)

We will refer to 1/πQ as the inverse match probability (IMP).
Consider first that Q is the major contributor to an LTDNA profile. Intuitively, if E implies that $GX=GQ$ then equality should be achieved in Eq. (3). The key idea of this paper is that if Hp is true then increasing numbers of LTDNA replicates should provide increasing evidence that $GX=GQ$, and so the ltLR should converge to the IMP. This holds even for mixtures if Q is the major contributor, since differential dropout rates should allow the alleles of Q to be identified from multiple replicates. However, any inadequacies in the underlying mathematical model or numerical approximations may become more pronounced with increasing numbers of replicates, preventing the ltLR from approaching the IMP. Therefore we propose to consider convergence of the ltLR towards the IMP as the number of replicates increases as an indicator of the validity of an algorithm to compute the ltLR when Q is the major contributor.
If Q is not the major contributor, even for many replicates there may remain ambiguity about the alleles of Q so that there remains a gap between the ltLR and IMP. However, the bound (3) still holds, and there is a useful guide to the appropriate value of the ltLR provided by the mixture LR for good-quality CSPs computed using only presence/absence of alleles [
• Weir B.S.
• Triggs C.M.
• Starling L.
• Stowell K.A.J.
• Buckleton J.
Interpreting DNA mixtures.
]. If under Hp the contributors are Q and U, where U denotes an unknown, unprofiled individual, and Hd corresponds to two unknown contributors X and U, an example of a mixture LR is
$mixLR=Pr(CSP=ABC,GQ=AB|Q,U)Pr(CSP=ABC,GQ=AB|X,U)=Pr(GUisoneofAC,BC,CC)Pr((GX,GU)isoneof(AA,BC),(AC,BB),(AB,CC),(AB,AC),(AB,BC),(AC,BC)),$
(4)

where within-pair ordering is ignored in the denominator. Under the standard population genetics model [
• Gill P.
• Brenner C.H.
• Buckleton J.S.
• Carracedo A.
• Krawczak M.
• Mayr W.R.
• Morling N.
• Prinz M.
• Schneider P.M.
• Weir B.S.
DNA commission of the International Society of Forensic Genetics: recommendations on the interpretation of mixtures.
,
• Gill P.
• Gusm ao L.
• Haned H.
• Mayr W.R.
• Morling N.
• Parson W.
• Prieto L.
• Prinz M.
• Schneider H.
• Schneider P.M.
• Weir B.S.
DNA commission of the International Society of Forensic Genetics: recommendations on the evaluation of STR typing results that may include drop-out and/or drop-in using probabilistic methods.
] and setting FST = 0, the mixLR for this example is
$Pr(CSP=ABC,GQ=AB|Q,U)Pr(CSP=ABC,GQ=AB|X,U)=2pA+2pB+pC6pApB(pA+pB+pC),$
(5)

where the p are population allele probabilities. As expected, mixLR < IMP = 1/2pApB. See Ref. [
• Balding D.J.
Weight-of-Evidence for Forensic DNA Profiles.
] for further details and examples. Note that the mixLR does not use peak height information.
Multiple LTDNA replicates should allow identification of all alleles present in any contributor, and hence the ltLR should reach the mixLR. In fact, ltLR will typically exceed mixLR because the alleles of different contributors may be distinguished over the multiple replicates through differential dropout rates. Indeed, Ref. [
• Ballantyne J.
• Hanson E.K.
• Perlin M.W.
DNA mixture genotyping by probabilistic computer interpretation of binomially-sampled laser captured cell populations: combining quantitative data for greater identification information.
] propose subsampling to generate different mixture ratios in low-template replicates as a strategy to assist mixture deconvolution. We cast light on this possibility below by considering a real CSP that has been profiled using multiple replicates at two different levels of sensitivity. More generally, we examine the behaviour of ltLR in relation to mixLR and IMP, and the utility of each of these for verifying the validity of ltLR computations.
likeLTD is an open-source R package that computes likelihoods for low-template DNA profiles [
• Balding D.J.
Evaluation of mixed-source, low-template DNA profiles in forensic science.
]. likeLTD allows for the designation of epg peaks as uncertain in addition to the usual allelic/non-allelic classification, but does not directly use epg peak heights. Uncertain alleles are treated as if they were masked in calculation of the likelihood: the presence/absence of the allele is regarded as unknown. The effect of an uncertain call on calculation of the likelihood is illustrated in Table 1. When B is called as uncertain rather than absent and the hypothesised contributor has a B allele, a dropout term D is removed from the likelihood because the dropout status of B is unknown. We use likeLTD here both to confirm its good performance in computing ltLRs, and to illustrate the value of the IMP as a strict upper bound and the mixLR as an approximate lower bound. We apply likeLTD to lab-based profiling replicates, simulated replicates, and replicates obtained by re-sampling the five actual replicates of a real CSP.
Table 1Likelihood calculations for a CSP when the queried contributor Q has genotype AB and [] indicates an allele designated as uncertain. Lp is the likelihood under the prosecution hypothesis, and D is the dropout probability. Under Hd are possible genotypes for the alternative contributor X, where Z is any other allele. Ld is the corresponding contribution to the likelihood under the defence hypothesis, where px is the probability of allele x, and D2 is the homozygote dropout probability.
CSPLpHdLd
AD(1 − D)AA$pA2(1−D2)$
AZ2pA(1 − pA)D(1 − D)
A[B]1 − DAA$pA2(1−D2)$
AB2pApB(1 − D)
AZ2pA(1 − pA − pB)D(1 − D)
Throughout this paper, ltLR, mixLR and IMP will be reported in units of bans, which is a base 10 logarithmic scale introduced as a measure of weight of evidence by Alan Turing during his wartime code breaking work [
• Irving J. Good
Studies in the history of probability and statistics. xxxvii am Turing's statistical work in world war ii.
]. Thus 6 bans corresponds to an LR of 1 million on the natural scale.

## 2. Materials and methods

### 2.1 Laboratory replicates

Cheek swab samples were obtained from five volunteers, and DNA was extracted using a PrepFiler Express BTA™ Forensic DNA Extraction Kit and the Life Technologies Automate Express™ Instrument as per the manufacturer's recommendations. The samples were then quantified using the Life Technologies Quantifiler® Human DNA Quantification kit as per the manufacturer's recommendations.
Each sample was serially diluted on a log 10 scale, and then amplified using the AmpFℓSTR® SGM Plus® PCR kit as per the manufacturer's recommendations on a Veriti® 96-Well Fast Thermal Cycler.
An ABI 3130 Sequencer was used to analyse 1 μL of the PCR products, with 10 second injections at 3 kV; these settings were used for all subsequent analyses. The results returned from the 3130 sequencer were analysed using GeneMapper® ID v3.2 to determine which samples were suitable for further use.
For the one-contributor investigation eight replicates of each of three conditions were created (Table 2). The conditions were created to investigate increasing dropout rate. For the 500 pg and 60 pg conditions, one-contributor hypotheses were compared, B under Hp and X under Hd, while for the 15 pg condition dropin was also modelled under both hypotheses (Table 3).
Table 2Sample preparation and genotyping protocol for all conditions examined in the lab-based experiments (described in Table 3). Each condition was replicated eight times. The initial DNA concentration (column 3), dilution (column 4) and volume (column 5) generate approximately the DNA mass indicated in column 6. Columns 7 and 8 show the number of PCR cycles and the volume of PCR product added to each well for the genotyping. Columns 9 and 10 show the ratio of Hi-Di™ formamide to GeneSan™ 400HD ROX™ and the volume of the mixture added to each well. Apmr stands for as per manufacturers recommendations.
ConditionContributorInit. conc. (ng μL−1)Dilution (%)Volume (μL)Mass (pg)CyclesProduct (μL)Formamide: ROXF/ROX mixture (μL)
(i)B31.011.650028apmrapmrapmr
(ii)B31.00.12.060
(iii)B31.00.015.015
(iv)A23.0117.650028apmrapmrapmr
C18.10.11630
(v)A23.00.122.46028apmrapmrapmr
C18.1122.0500
(vi)A23.00.12.76028apmrapmrapmr
B31.00.12.060
C18.10.13.560
(vii)A23.00.12.760281600:19
B31.00.12.060
C18.10.13.560
(viii)A23.00.12.760289366:111
B31.00.12.060
C18.10.13.560
(ix)A23.00.12.76030apmrapmrapmr
B31.00.12.060
C18.10.13.560
Table 3Experimental conditions and hypotheses compared. pg denotes picograms and measures DNA mass; Pr(D) denotes the probability of dropout for a heterozygote allele, while Pr(C) denotes the probability of dropin. Pr(unc) indicates the probability of designating a CSP allele as uncertain. υ indicates the number of uncertain dropins per locus per replicate; see text for further details of “Condition”. Q denotes the queried contributor, who is one of A, B or C as indicated in parentheses. X is an unknown alternative to Q under Hd, while U1 and U2 are unknown contributors under both Hp and Hd.
Study# ContributorsConditionHpHd
Lab-based1500 pg (i)Q (B)X
60 pg (ii)Q (B)X
15 pg (iii)Q (B) + dropinX + dropin
2A=500 pg; C=30 pg (iv)Q (A) + dropinX + dropin
Q (A) + U1X + U1
Q (C) + U1X + U1
A=60 pg; C=500 pg (v)Q (C) + dropinX + dropin
Q (C) + U1X + U1
Q (A) + U1X + U1
328 cycles (vi)Q (A) + U1 + U2X + U1 + U2
Phase 1 (vii)Q (A) + U1 + U2X + U1 + U2
Phase 2 (viii)Q (A) + U1 + U2X + U1 + U2
30 cycles (ix)Q (A) + U1 + U2X + U1 + U2
Simulation1PrB(D) = 0; Pr(C) = 0Q (B)X
PrB(D) = 0.4; Pr(C) = 0.05Q (B) + dropinX + dropin
PrB(D) = 0.8; Pr(C) = 0.05Q (B) + dropinX + dropin
Pr(unc) = 0.8; υ∼ Pois(λ = 1)Q (B)X
Pr(unc) = 0.4; υ∼ Pois(λ = 1)Q (B)X
2PrA,C(D) = {0.2, 0.8}; Pr(C) = 0Q (A) + dropinX + dropin
Q (A) + U1X + U1
Q (C) + U1X + U1
PrA,C(D) = {0.2, 0.6}; Pr(C) = 0Q (A) + dropinX + dropin
Q (A) + U1X + U1
Q (C) + U1X + U1
3PrA,B,C(D) = {0.8,0.5,0.2}; Pr(C) = 0Q (A) + U1 + U2X + U1 + U2
PrA,B,C(D) = {0.5,0.5,0.5}; Pr(C) = 0Q (A) + U1 + U2X + U1 + U2
PrA,B,C(D) = {0.2,0.5,0.8}; Pr(C) = 0Q (A) + U1 + U2X + U1 + U2
Real-world≥3Standard and sensitiveQ + U1 + U2X + U1 + U2
Standard onlyQ + U1 + U2X + U1 + U2
Sensitive onlyQ + U1 + U2X + U1 + U2
For the two-contributor investigation eight replicates of each of two conditions were created (Table 2). The major and minor contributors were reversed between conditions, with an increased DNA contribution from the minor. These samples were amplified and analysed as described previously. Two-contributor hypotheses were compared, with each of A and C in turn playing the role of Q, while the other contributor was treated as unknown. Additionally one-contributor-plus-dropin hypotheses were compared, with only the major contributor playing the role of Q (Table 3).
For the three-contributor investigation eight replicates of each of four conditions were created (Table 2). The conditions were created to investigate different profiling protocols. The Phase 1 and Phase 2 conditions are post-PCR purification protocols designed to enhance the sensitivity of detection of the standard protocol [
• Roeder A.D.
• Elsmore P.
• Greenhalgh M.
• McDonald A.
Maximizing DNA profiling success from sub-optimal quantities of DNA: a staged approach.
], and both involve concentrating the post-PCR product using an Amicon® PCR microcon unit according to the manufacturer's recommendations. Phase 1 enhancement increases the amount of formamide in the mixture compared to the manufacturer's recommendations, while Phase 2 enhancement increases the amount of DNA, formamide and ROX compared to Phase 1. For all four conditions (30 cycles, 28 cycles, Phase 1, and Phase 2), three-contributor hypotheses were compared, with A playing the role of Q and the other contributors treated as unknown (Table 3). Dropin was not modelled under either hypothesis, although dropin was included in the simulations. This reflects a realistic challenge for few replicates with multiple contributors, whereby any dropin alleles may be wrongly attributed to one of the contributors. However the incorrect model will lead to deterioration of inferences for larger numbers of replicates.

### 2.2 Simulated replicates

All of the conditions that we now describe were simulated in eight replicates, with the whole simulation being performed five times. Initially a number of single-contributor CSPs were simulated using the profile of individual B. The first condition investigated was a “perfect match”, in which all eight replicates generated exactly the profile of B. Next, we introduced mild dropout (Pr(D) = 0.4) and severe dropout (Pr(D) = 0.8) of the alleles of B, in each case with dropins included at rate Pr(C) = 0.05 (at most one dropin per locus per replicate). The homozygous dropout probability was set equal to Pr(D)2/2, as suggested by [
• Balding D.J.
• Buckleton J.
Interpreting low template DNA profiles.
]. We then examined the effect of uncertain allele designations by randomly designating some alleles of B as uncertain, first with Pr(unc) = 0.4 and then Pr(unc) = 0.8. In both conditions, at each locus and in each replicate a Poisson mean one number of alleles not in the profile of B was also designated as uncertain, with types randomly selected according to frequencies in the UK Caucasian database. For all these simulated profiles, one-contributor hypotheses were compared, B under Hp and X under Hd.
Next two-contributor CSPs were simulated, based on the profiles of A and C. Two conditions were simulated, both used PrA(D) = 0.2, while PrC(D) was initially 0.8 and then 0.6. Dropin was not simulated. For shared alleles the dropout probability was the product of the dropout probabilities for each contributor having that allele. Two-contributor hypotheses were compared, with each of A and C in turn taking the role of Q, while the other was treated as unknown in the analysis. Additionally one-contributor-plus-dropin hypotheses were compared, only for A playing the role of Q (Table 3).
Three-contributor CSPs were then simulated under three conditions, with dropout probabilities for Donors A, B and C as shown in Table 3. Dropin was included as for the one-contributor simulations. Three-contributor hypotheses were compared, with A playing the role of Q and the other two contributors being treated as unknown.

### 2.3 Crime case replicates

We used a CSP from an actual crime investigation, consisting of five replicates: two using standard SGM+ profiling and three generated using an LCN protocol with 34 PCR cycles (Table 4). This example was submitted to us for likeLTD analysis, and as is typical only limited information about the profiling protocol was provided by the profiling lab. These details are not required by likeLTD because it estimates the unknown parameters from the CSP allele designations. We re-sampled the five actual replicates to generate simulated profiles with up to eight replicates, consisting of standard replicates only, sensitive replicates only, or both. Six distinct alleles were observed at locus D8, but no more than three replicated alleles were observed at any locus. Three-contributor hypotheses were compared, with all contributors unknown under Hd, and no dropin (Table 3).
Table 4Five replicates of a crime scene profile, three from a sensitive LTDNA profiling technique and two from standard DNA profiling. Alleles shown in [] were called as uncertain.
LocusSensitive profilingStandard profiling
Run 1Run 2Run 3Run 4Run 5
D316, [15]16, [15]16, 18, [15]1616
vWA15, 16, [17]15, [14]15, 18, [14]1515
D16999, 11, [10]99
D217, 19, 2416, 17, 24,[23]17, [16]2424
D88, 13, 15, 168, 12, 13, 16, [15]8, 13, 14, 16, [15][8]
D2130, 32, 33.232, 32.2, 33.232, 32.2, 33.2, 34, [31][32], [32.2][33.2]
D1812, 1712, 17, 1912, 17, [11], [16][17]17
D1914, 21, [13]11, 14, [13]14, [13]1414
TH016, 9.36, 9.36, 8, 9.3[6], [9.3][6]
FGA2121, [20]21, 2021

## 3. Results

### 3.1 One contributor

#### 3.1.1 Lab-based

For the good-template experiments (500 pg), Fig. 1 (left) shows that the ltLR equals the IMP for all numbers of replicates (one through eight). This is the expected result, and the exercise shows that in this simple setting there is no deterioration in the quality of the computed LR for large numbers of replicates. Low DNA template (60 pg) generates an ltLR about 1.6 bans below the IMP for one replicate, but the gap is very small for two replicates and is negligible for larger numbers of replicates. For very low DNA template (15 pg) the ltLR is just under 6 bans for a single replicate, about 6 bans below the IMP. Replicate profiling substantially narrows the gap, but does not completely close it, with a difference of about 3 decibans remaining at eight replicates.

#### 3.1.2 Simulation

The corresponding simulation studies show broadly similar trends to the lab-based data. For both the perfect match (Pr(D) = 0) and mild dropout (Pr(D) = 0.4) conditions, the median ltLR rapidly reaches the IMP but does not exceed it, while under severe dropout (Pr(D) = 0.8) the median ltLR rises towards the IMP but does not reach it (Fig. 1, middle). For the low and high rates of uncertain calls, the IMP is approximately reached at a five and eight replicates, respectively (Fig. 1, right).

### 3.2 Two contributors

#### 3.2.1 Lab-based

When the minor contributor provides only 30 pg of DNA (Fig. 2, top left panel), then if Q is the major contributor the ltLR is very close to the IMP for all numbers of replicates, whereas if Q is the minor contributor then there remains a substantial gap between ltLR and IMP even at eight replicates. However, even with this very low template, the ltLR exceeds the mixLR beyond five replicates. When the major and minor contributors are reversed, and the amount of DNA from the minor is doubled (Fig. 2, bottom left), then if Q is the minor contributor the ltLR substantially exceeds mixLR from six replicates and rises to within two bans of the IMP at eight replicates. Under both conditions, the two-contributor analysis gives a very similar result to the one-contributor-with-dropin analysis.

#### 3.2.2 Simulation

When the minor contributor is subject to high dropout (Fig. 2, top right), then if Q is the major contributor the ltLR exceeds the mixLR after one replicate, and rises rapidly to within about 2 bans of the IMP, but the gap narrows only slowly thereafter. The one-contributor-plus-dropin analysis gives an ltLR that is broadly similar to the two contributor analysis, but with a wider range indicating greater variability. If Q is the minor contributor, the median ltLR increases rapidly from a low base, and appears to stabilise after about five replicates, about four bans below the IMP but exceeding the mixLR. The range increases after three replicates, and remains high up to eight replicates.
With reduced dropout for the minor contributor (Fig. 2, bottom right), inferring the presence of a major contributor Q is harder because of additional masking by the minor contributor. The median ltLR in both the two contributor and one-contributor-plus-dropin analyses eventually reaches within 2 bans of the IMP, with the latter showing a greater range. Conversely, the lower dropout rate leads to improved inference for a minor contributor Q, with the median ltLR rising to about three bans below the IMP at eight replicates, and exceeding the mixLR from four replicates. Interestingly, after six replicates the range of the minor contributor ltLR overlaps the range for the major contributor.

### 3.3 Three contributors

#### 3.3.1 Lab-based

The 30 PCR cycles condition gives the highest ltLR at one replicate but little improvement with additional replicates (Fig. 3, left). The other amplification methods do show an increasing ltLR trend with additional replicates, but in no case did the ltLR reach within four bans of the IMP. As expected, the ltLR for both phase 1 and phase 2 enhancement exceeds that for standard 28 PCR cycles at all numbers of replicates, and phase 2 enhancement ltLR typically gives a small improvement over phase 1 enhancement. For 30 PCR cycles, the ltLR exceeds the mixLR for a single replicate but dips slightly below it at six replicates. For the other conditions, the mixLR is always exceeded from four replicates.

#### 3.3.2 Simulation

All three curves in Fig. 3 (middle) show an increasing trend with number of replicates, with the median ltLR being in the expected order throughout (decreasing ltLR with increasing dropout for Q). The median ltLR exceeds the mixLR after one replicate (low dropout), after two replicates (medium dropout) and after four replicates (high dropout). The range is often wide, reflecting a strong dependence of the ltLR on the details of the simulation (in particular the number of alleles shared across contributors).

#### 3.3.3 Real-world case

The ltLR returned when only standard or only sensitive replicates are used shows a similar trend, but nearly five bans lower for the standard replicates (Fig. 3, right). For three or more replicates, using mixed types of replicates is superior even to only using sensitive replicates, coming to within two bans of the IMP. This partly reflects the limited pool of replicates used in the actual crime case, but suggests that using different sensitivities in the profiling replicates may convey an advantage due to different contributors being better distinguished.

## 4. Discussion

We have shown that ltLR computed by likeLTD is bounded above by the IMP in every condition considered, as predicted by theory (Eq. (3)). That the bound is often tight when Q is the major contributor (Fig. 1, Fig. 2 (top)) supports the validity of the underlying mathematical model, and its correct implementation in the likeLTD software. Our results should help counter any misconception that combining multiple noisy profiling replicates only compounds the noise: in fact, multiple noisy replicates can fully recover the genotype of a contributor [
• Schneps L.
• Colmez C.
Math on trial: how numbers get used and abused in the courtroom.
].
A novel feature of likeLTD, is that it can accommodate uncertain allele designations, which diminishes the problem of an all-or-nothing allele call, therefore mitigating the problem highlighted by [
• Budowle B.
• Eisenberg A.J.
• van Daal A.
Validity of low copy number typing and applications to forensic science.
] of choosing a detection threshold. We have shown (Fig. 1 (right)) that introducing many uncertain allele calls leads to ltLRs that satisfy the bound, which is reasonably tight with as few as three replicates even when 80% of true alleles are designated as uncertain and there are also multiple uncertain non-alleles.
We have further shown that mixLR, the LR computed from knowing every allele that is represented in the profile of at least one contributor to the CSP, is often surpassed after only a handful of replicates. Then, multiple LTDNA replicates provide stronger evidence than a single good quality profile correctly representing the alleles of all contributors, which occurs because the alleles of different contributors can to some extent be distinguished through differential dropout rates in multiple replicates. These results lend support in principle to the proposal of [
• Ballantyne J.
• Hanson E.K.
• Perlin M.W.
DNA mixture genotyping by probabilistic computer interpretation of binomially-sampled laser captured cell populations: combining quantitative data for greater identification information.
].
Fig. 2 shows that, for two-person mixtures, the analysis assuming one-contributor-plus-dropin gave a very good approximation for the lab-based replicates (left panels), and a reasonably good approximation for the simulation replicates, but with more variable ltLR values, as indicated by the wider range.

### 4.1 Choice of profiling technique

We generated three-contributor CSPs in order to compare different LTDNA profiling techniques. We chose the most challenging condition in which all three contribute the same DNA template, making it impossible to deconvolve the mixture into the genotypes of individual contributors. We found that PCR performed with 28 cycles (regardless of enhancement) is preferable to 30 cycle PCR beyond one replicate (Fig. 3). More PCR cycles introduces more stochasticity in the results, as stated in the AmpFℓSTR® SGM Plus® PCR Amplification Kit user guide. We found that enhancement of the post-PCR sample is advantageous, with Phase 2 enhancement providing a small further improvement over Phase 1 (Fig. 3). These results support those of Forster et al. [
• Forster L.
• Thomson J.
• Kutranov S.
Direct comparison of post-28-cycle PCR purification and modified capillary electrophoresis methods with the 34-cycle “low copy number”(LCN) method for analysis of trace forensic DNA samples.
], who demonstrated that increasing PCR cycles increases the size of stutter peaks and the incidence of dropin; we observed no improvement in the WoE for 30 PCR cycles, possibly due to these stochastic effects.
The results from the real crime case (Fig. 3, right) suggest that if possible, a mixture of LTDNA replicates with differing sensitivities should be employed, as this allows better discrimination between the alleles of different contributors and hence a higher ltLR than the same number of replicates all using the same sensitivity.

### 4.2 Use of replicates

Splitting the sample reduces the quality of results expected in each replicate compared with that which would be obtained from a single profiling run using all available DNA. Grisedale and van Daal [
• Grisedale K.S.
• van Daal A.
Comparison of STR profiling from low template DNA extracts with and without the consensus profiling method.
] favour use of a single run, but their comparison was with a consensus sequence obtained from multiple replicates, rather than the more efficient statistical analysis available through analysing individual replicates. Our results show increasing information obtained from additional replicates, which may tilt the argument towards use of multiple replicates but we have not done a comparison directly addressing this question. To fully test the performance of likeLTD in relation to mixLR and IMP we have used up to eight replicates. Taberlet et al. [
• Taberlet P.
• Griffin S.
• Goossens B.
• Questiau S.
• Manceau V.
• Escaravage N.
• Waits L.P.
• Bouvet J.
Reliable genotyping of samples with very low DNA quantities using PCR.
] suggest seven replicates to generate a quality profile when the amount of DNA is low, but this many replicates is rarely available for low-template crime samples [
• Budowle B.
• Eisenberg A.J.
• van Daal A.
Validity of low copy number typing and applications to forensic science.
].

## Acknowledgements

CDS is funded by a PhD studentship from the UK Biotechnology and Biological Sciences Research Council and Cellmark Forensic Services.

## References

• Steele C.D.
• Balding D.J.
Statistical evaluation of forensic DNA profile evidence.
Annu. Rev. Stat. Appl. 2014; 1: 20-21https://doi.org/10.1146/annurev-statistics-022513-115602
• Curran J.M.
• Gill P.
• Bill M.R.
Interpretation of repeat measurement DNA evidence allowing for multiple contributors and population substructure.
Forensic Sci. Int. 2005; 148: 47
• Gill P.
• Haned H.
A new methodological framework to interpret complex DNA profiles using likelihood ratios.
Forensic Sci. Int. Genet. 2012; https://doi.org/10.1016/j.fsigen.2012.11.002
• Cowell R.G.
• Graversen T.
• Lauritzen S.
• Mortera J.
Analysis of DNA mixtures with artefacts.
in: ArXiv e-Prints. 2013, February
• Weir B.S.
• Triggs C.M.
• Starling L.
• Stowell K.A.J.
• Buckleton J.
Interpreting DNA mixtures.
J. Forensic Sci. 1997; 42: 213-222
• Gill P.
• Brenner C.H.
• Buckleton J.S.
• Carracedo A.
• Krawczak M.
• Mayr W.R.
• Morling N.
• Prinz M.
• Schneider P.M.
• Weir B.S.
DNA commission of the International Society of Forensic Genetics: recommendations on the interpretation of mixtures.
Forensic Sci. Int. 2006; 160: 90-101
• Gill P.
• Gusm ao L.
• Haned H.
• Mayr W.R.
• Morling N.
• Parson W.
• Prieto L.
• Prinz M.
• Schneider H.
• Schneider P.M.
• Weir B.S.
DNA commission of the International Society of Forensic Genetics: recommendations on the evaluation of STR typing results that may include drop-out and/or drop-in using probabilistic methods.
Forensic Sci. Int. Genet. 2012; https://doi.org/10.1016/j.fsigen.2012.06.002
• Balding D.J.
Weight-of-Evidence for Forensic DNA Profiles.
Wiley, 2005
• Ballantyne J.
• Hanson E.K.
• Perlin M.W.
DNA mixture genotyping by probabilistic computer interpretation of binomially-sampled laser captured cell populations: combining quantitative data for greater identification information.
Sci. Justice. 2013; 53 (ISSN 1355-0306): 103-114https://doi.org/10.1016/j.scijus.2012.04.004
• Balding D.J.
Evaluation of mixed-source, low-template DNA profiles in forensic science.
Proc. Natl. Acad. Sci. U. S. A. 2013; https://doi.org/10.1073/pnas.1219739110
• Irving J. Good
Studies in the history of probability and statistics. xxxvii am Turing's statistical work in world war ii.
Biometrika. 1979; 66: 393-396
• Roeder A.D.
• Elsmore P.
• Greenhalgh M.
• McDonald A.
Maximizing DNA profiling success from sub-optimal quantities of DNA: a staged approach.
Forensic Sci. Int. Genet. 2009; 3: 128-137https://doi.org/10.1016/j.fsigen.2008.12.004
• Balding D.J.
• Buckleton J.
Interpreting low template DNA profiles.
Forensic Sci. Int. Genet. 2009; 4: 1-10https://doi.org/10.1016/j.fsigen.2009.03.003
• Schneps L.
• Colmez C.
Math on trial: how numbers get used and abused in the courtroom.
Wiley Online Library, 2013
• Budowle B.
• Eisenberg A.J.
• van Daal A.
Validity of low copy number typing and applications to forensic science.
Croat. Med. J. 2009; 50: 207-217https://doi.org/10.3325/cmj.2009.50.207
• Forster L.
• Thomson J.
• Kutranov S.
Direct comparison of post-28-cycle PCR purification and modified capillary electrophoresis methods with the 34-cycle “low copy number”(LCN) method for analysis of trace forensic DNA samples.
Forensic Sci. Int. Genet. 2008; 2: 318-328https://doi.org/10.1016/j.fsigen.2008.04.005
• Grisedale K.S.
• van Daal A.
Comparison of STR profiling from low template DNA extracts with and without the consensus profiling method.
Invest. Genet. 2012; 3: 14https://doi.org/10.1186/2041-2223-3-14
• Taberlet P.
• Griffin S.
• Goossens B.
• Questiau S.
• Manceau V.
• Escaravage N.
• Waits L.P.
• Bouvet J.
Reliable genotyping of samples with very low DNA quantities using PCR.
Nucleic Acids Res. 1996; 24: 3189-3194


text:
### Winner of Caption Contest 13 ' You think you are Funny?'

The winner of the Caption Contest 13 ‘You think you are funny?’
1) Rajlakshmi: 'The H(L)appy Family'

Congratulations!!!

Drop me a mail at and I will mail you the trophy.
Hope to see you in another round of Caption Contest this Wednesday.

1. wooaaaa i don't believe it :D finally i managed to win :D
thanks :D

2. Congrats to the winner! This picture is quite challenging but the caption fits it so nicely :)


text:
RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
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 Mat. Sb.: Year: Volume: Issue: Page: Find

 Mat. Sb., 2014, Volume 205, Number 7, Pages 3–24 (Mi msb8268)

$N^\pm$-integrals and boundary values of Cauchy-type integrals of finite measures

R. A. Aliyev

Baku State University

Abstract: Let $\Gamma$ be a simple closed Lyapunov contour with finite complex measure $\nu$, and let $G^+$ be the bounded and $G^-$ the unbounded domains with boundary $\Gamma$. Using new notions (so-called $N$-integration and $N^+$- and $N^-$-integrals), we prove that the Cauchy-type integrals $F^+(z)$, $z\in G^+$, and $F^-(z)$, $z\in G^-$, of $\nu$ are Cauchy $N^+$- and $N^-$-integrals, respectively. In the proof of the corresponding results, the additivity property and the validity of the change-of-variable formula for the $N^+$- and $N^-$-integrals play an essential role.
Bibliography: 21 titles.

Keywords: finite complex Borel measure, Cauchy-type integral, nontangential boundary values, Cauchy integral, $Q$-integral, $Q'$-integral, $N$-integration.

DOI: https://doi.org/10.4213/sm8268

Full text: PDF file (570 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2014, 205:7, 913–935

Bibliographic databases:

UDC: 517.518.234+517.547.73
MSC: Primary 28A25; Secondary 26A42, 42B25

Citation: R. A. Aliyev, “$N^\pm$-integrals and boundary values of Cauchy-type integrals of finite measures”, Mat. Sb., 205:7 (2014), 3–24; Sb. Math., 205:7 (2014), 913–935

Citation in format AMSBIB
\Bibitem{Ali14} \by R.~A.~Aliyev \paper $N^\pm$-integrals and boundary values of Cauchy-type integrals of finite measures \jour Mat. Sb. \yr 2014 \vol 205 \issue 7 \pages 3--24 \mathnet{http://mi.mathnet.ru/msb8268} \crossref{https://doi.org/10.4213/sm8268} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3242643} \zmath{https://zbmath.org/?q=an:06381823} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2014SbMat.205..913A} \elib{http://elibrary.ru/item.asp?id=21826633} \transl \jour Sb. Math. \yr 2014 \vol 205 \issue 7 \pages 913--935 \crossref{https://doi.org/10.1070/SM2014v205n07ABEH004403} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000344080400001} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84908472736}

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• https://doi.org/10.4213/sm8268
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. R. A. Aliev, “On Taylor coefficients of Cauchy-type integrals of finite complex measures”, Complex Var. Elliptic Equ., 60:12 (2015), 1727–1738
2. “On the Properties of Q- and Q `-Integrals of the Function Measurable on the Real Axis”, Proc. Inst. Math. Mech., 41:1 (2015), 56–62
3. R. A. Aliev, “Riesz's equality for the Hilbert transform of the finite complex measures”, Azerb. J. Math., 6:1 (2016), 126–135
4. R. A. Aliev, “On properties of Hilbert transform of finite complex measures”, Complex Anal. Oper. Theory, 10:1 (2016), 171–185
5. R. A. Aliev, “On Laurent coefficients of Cauchy type integrals of finite complex measures”, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb., 42:2 (2016), 292–303
6. R. A. Aliev, “Representability of Cauchy-type integrals of finite complex measures on the real axis in terms of their boundary values”, Complex Var. Elliptic Equ., 62:4 (2017), 536–553
7. R. A. Aliev, Kh. I. Nebiyeva, “The $A$-integral and restricted Ahlfors–Beurling transform”, Integral Transform. Spec. Funct., 29:10 (2018), 820–830
8. Aliev R.A., Amrahova A.F., “Properties of the Discrete Hilbert Transform”, Complex Anal. Oper. Theory, 13:8 (2019), 3883–3897
9. Aliev R.A., Nebiyeva I Kh., “The a-Integral and Restricted Complex Riesz Transform”, Azerbaijan J. Math., 10:1 (2020), 209–221
•  Number of views: This page: 951 Full text: 79 References: 60 First page: 66


text:
# mathlibdocumentation

category_theory.pempty

# The empty category

Defines a category structure on pempty, and the unique functor pempty ⥤ C for any category C.

def category_theory.functor.empty (C : Type u)  :

The canonical functor out of the empty category.

Equations
def category_theory.functor.empty_ext {C : Type u} (F G : C) :
F G

Any two functors out of the empty category are isomorphic.

Equations

Any functor out of the empty category is isomorphic to the canonical functor from the empty category.

Equations
theorem category_theory.functor.empty_ext' {C : Type u} (F G : C) :
F = G

Any two functors out of the empty category are equal. You probably want to use empty_ext instead of this.


text:
Page History

Key

• This line was removed.
• Formatting was changed.

...

Cast to void can be used to explicitly mark a variable unused, suppressing the warnings from the compiler or the static analyzer: (void) unused_variable;.

Native types

Usage of intunsigned, and all other native integral types is not permitted. Use cstdint instead.

Usage of floatdoublelong double is not recommended; consider aliasing them to float32_t and such instead.

Scoping

The statement using namespace should not be used, except in a very local scope (see an example below). Usage of this statement in a global scope or in a namespace scope is strictly prohibited.

...


text:
# Solution needed for a physics problem

Hi brilliant physics problem solvers, I have expirenced a problem with this question and can anyone of the 38 problem solvers of the question Rolling down the stairs give a solution

Note by Kahsay Merkeb
5 years, 5 months ago

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
• Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
• Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.

MarkdownAppears as
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# up as a code block.

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# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting.
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2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$

Sort by:

I agree, it would be great to see a solution.

Staff - 5 years, 5 months ago

Refer to those people @Suyeon Khim ,@Peter Taylor ,@Vishnuram Leonardodavinci @Thaddeus Abiy @Beakal Tiliksew .I also couldnot solve the problem though I got it after a week

- 5 years, 5 months ago

i have posted a solution, u could check it out. And sorry for the late response, i just had no free time.

- 5 years, 5 months ago

o

noooooooooo

- 5 years, 5 months ago


text:
# using macros within a (new) environment [closed]

I'm working on my CV, and am trying to format job/education listings to look something like follows:

Company or University (Location)
Some description of what I did there.

I'm putting enough of these in that I'd like to define an environment to handle the formatting so I don't have to hard-code it (and/or have to re-do every single one individually when I change my mind!). I.e. I'd like to be able to do something like

\begin{job}
\institution{Company or University}
\location{Location}
\description{Some description of what I did there.}
\end{job}

And have it output the above. I kind of blindly tried to define new commands for \institution, \location, \position, etc. using e.g.

\def\institution#1{\gdef{\@institution}{#1}}

and then create a new environment that does the formatting in the "after environment" code, like so:

\newenvironment{job}{}{\textbf{\@institution}}

I put both of these inside a \makeatletter...\makeatother block, and my document compiles fine with this in the preamble. But as soon as I try to actually use this environment, e.g. insert

\begin{job}
\institution{Someplace I worked}
\end{job}

somewhere in the document body, I get an error that says

Missing control sequence inserted: <inserted text>
\inaccessible

when it reads the \institution{Someplace I worked} line and a second that says

Undefined control sequence: <recently read> \@institution

when it reads the \end{job} line.

-

## closed as too localized by Marco Daniel, Joseph Wright♦Aug 11 '12 at 9:10

This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

\gdef{\@institution}{#1} is a syntax error you can not put braces around the command being defined with the primitive def syntax. \gdef\@institution{#1}} – David Carlisle Aug 10 '12 at 18:54
Also, there is no need to define those commands as global. It might be a better idea to define local commands in the "begin" portion of \newenvironment – mrf Aug 10 '12 at 21:49
Ah! Thanks, @david-carlisle - the syntax error was the culprit. It worked once I fixed that. – Jennifer L Aug 10 '12 at 22:44
@JenniferL Is it OK for you if we close this question since it seems that you have fixed the issue? – percusse Aug 10 '12 at 23:01
Yes - I was just about to update the question with the solution, but it's fine to close it. Thanks for the help! – Jennifer L Aug 10 '12 at 23:09

Here's a couple of ideas- one is using a \newcommand, and one uses a \newenvironment; perhaps you can tweak them as you see fit

Code:

\documentclass{article}

% as a command
\newcommand{\job}[4]{% 1: company, 2: location, 3: title, 4: description
\noindent\begin{tabular}{@{}l}%
\bfseries{#1 (#2)}\\
\itshape{#3}\\
#4
\end{tabular}%
}

% or perhaps an environment
\newenvironment{myjob}[3]{% 1: company, 2: location, 3: title
\noindent\begin{tabular}{@{}p{\textwidth}@{}}%
\bfseries{#1 (#2)}\\
\itshape{#3}\\
}%
{%
\end{tabular}%
}

\begin{document}

\job{Company or University}%
{Location}%
{Some description of what I did there.}

\begin{myjob}{Another institution}%
{Somewhere else}%
{Another thing}%
Long description of stuff| could be anything
\begin{itemize}
\item discovered
\item a
\item few
\item important
\item things
\end{itemize}
\end{myjob}

\end{document}

-


text:
# Moment of inertia of a solid sphere; spot the mistake [closed]

What an I doing wrong?:

$$I = \int r^2dm$$

$$M = \rho \frac{4}{3} \pi r³$$

$$dM = \rho \frac{4}{3} \pi 3r²dr = \rho 4 \pi r²dr$$

$$I = \rho 4π \int r^4dr = \rho 4π \frac{r^5}{5} = \frac{3Mr^2}{5} = \frac{3}{5} Mr^2$$

Instead of factor $$\frac{2}{5}$$.

• spot the mistake ... What an I doing wrong? Please be aware that check-my-work are usually considered off-topic on this site. I think the members who answered made an exception for your question because you made a conceptual error rather than a calculational one. – G. Smith Mar 24 at 2:46
• Yes I'm aware that this isn't really well defined for this sites but i knew my question was a conceptual error and since my question is about the method of calculation I must provide my work thus far. But I'll definitely try to avoid questions like these. – Pim Laeven Mar 26 at 23:15

You have not used the correct definition of $$I$$.

The moment of inertia is defined as $$I=\int r^2_\perp dm=\int r^2_\perp\rho dV.$$ Using spherical coordinates $$\vec r=\begin{pmatrix}x\\y\\z\end{pmatrix}=r\begin{pmatrix}\cos\varphi\sin\theta\\ \sin\varphi\sin\theta\\\cos\theta\end{pmatrix},$$ we get $$r_\perp^2=x^2+y^2=r^2\sin^2\theta\,\,\,\text{and}\,\,\,dV=r^2\sin\theta drd\varphi d\theta$$ and thus $$I=\rho\int r^2\sin^2\theta dV=\rho\int_0^R r^4dr\int_0^{2\pi} d\varphi\int_0^\pi\sin^3\theta d\theta=\rho\frac{R^5}{5}2\pi\frac{4}{3}=\frac{2}{5}MR^2$$since$$\rho=\frac{M}{4/3\pi R^3}.$$

You need to evaluate the moment of inertia about an axis through the center, and that distance is not always $$r$$ for a mass element $$dm$$ from the axis.

• Ah of course, how stupid of me, thanks a lot! – Pim Laeven Mar 23 at 23:12
• You are welcome. – John Darby Mar 24 at 12:34

While your calculation is correct, your derivation starts from the wrong premise.

A moment of inertia is calculated about a given axis of rotation, not about a point.

I assume in your case the axis of rotation runs through the Centre of Mass of the uniform sphere.

In that case, you need to slice up the sphere into infinitesimal cylinders (circular plates) along the axis, then sum by integration the moments of inertia of these plates along the axis.

You can find the full derivation here.