domain: math shs

text:
Aoife O'brien
email: a.obrien47@nuigalway.ie

Máire Ní Ríordáin
email: maire.niriordain@ucc.ie

Examining difficulties in initial algebra: Prerequisite and algebra content areas for Irish post-primary students

Keywords: Algebra, diagnostic test, secondary education

This research aims to investigate the algebraic under performance of second year post-primary students in Ireland (approximate age 14 years). To this end a diagnostic test for algebra has been developed to profile and identify students who are struggling with algebra. This paper examines the development of the test, which involved the identification of key mathematical content areas that are critical for success in algebra. Both prerequisite, and algebra content areas are key to a students' success in algebra and how each of these areas contribute to a students' progress with algebra is discussed in this theoretical paper. Test items have been selected and adapted from the literature which are aligned with both the key content areas and the Irish mathematics syllabus at junior cycle, the initial three years of post-primary education in Ireland.

Introduction

This paper reports on the development of a diagnostic test for algebra, designed specifically for second year post-primary students in Ireland. It is noted in the literature that few adequate assessments are available to provide formative information on a students' progress with algebra, but they are essential to allow timely and informed instructional decisions for teachers [START_REF] Ketterlin-Geller | Measuring Middle School Students' Algebra Readiness: Examining Validity Evidence for Three Experimental Measures[END_REF]. A key outcome of this research will be a validated diagnostic test specifically designed to identify students' conceptual errors when working with algebra and aligned with the Irish post-primary mathematics syllabus. The test can be used by teachers in a normal class period (35-40 minutes as relevant to the Irish context) to help inform their instruction of algebra. This paper focuses on identifying key content areas which are required for success in algebra and accordingly developing a bank of questions to support the development of a diagnostic test.

Background and rationale

The mathematical deficiencies of students entering third level education in Ireland and internationally is widely reported and commonly referred to as the "Maths Problem" [START_REF] Treacy | Trends in basic mathematical competencies of beginning undergraduates in Ireland, 2003-2013[END_REF]. Over a decade ago it had become apparent that there were issues with the mathematics curriculum at post-primary level in Ireland resulting in radical reform of the syllabi, introduced as "Project Maths" [START_REF] Treacy | Trends in basic mathematical competencies of beginning undergraduates in Ireland, 2003-2013[END_REF]. The vast majority of students study mathematics throughout their time in post-primary education and the syllabus is delivered in five interwoven strands; 1. Probability and Statistics,2. Trigonometry and Geometry,3. Number,4. Algebra and 5. Functions. Mathematics is offered at higher, ordinary and foundation levels throughout. Despite the changes made to the Irish syllabus and teaching methods, it is clear that problems with algebra persist. This is evidenced in the latest chief examiners report which states students need to "gain comfort and accuracy in the basic skills of computation, algebraic manipulation and calculus" (Department of Education and Skills, 2015 p. 29). Algebra serves as a gateway to higher mathematics and deficiencies in basic algebra result in overall mathematical deficiencies in students [START_REF] Lawson | What can we expect from A-Level Mathematics Students[END_REF]. However, issues with the teaching and learning of algebra remain internationally, and research into progress measuring instruments to inform the area continue [START_REF] Ketterlin-Geller | Measuring Middle School Students' Algebra Readiness: Examining Validity Evidence for Three Experimental Measures[END_REF]. Given the poor performance of Irish students in algebra a diagnostic test has been developed to give insight into how second year students perform in this area. It is noted that many tests exist to measure performance in algebra and these provide useful information about the content areas in which students need further assistance. However, few of these tests provide information about why students are struggling [START_REF] Russell | Diagnosing students' misconceptions in algebra: Results from an experimental pilot study[END_REF]. This test may help inform part of why students are struggling by looking at the conceptual errors students make in other content areas. These content areas, which include fractions, equality and patterns for example, have been identified in the literature as essential for the understanding of algebra [START_REF] Bush | Prerequisite algebra skills and associated misconceptions of middle grade students: A review[END_REF][START_REF] Warren | Patterns that support early algebraic thinking in the elementary school[END_REF].

Theoretical frame

The theoretical framework guiding this research is that the conceptual errors students make in working with algebra perpetuate the difficulties they encounter, subsequently interfering with their understanding of various algebraic concepts [START_REF] Russell | Diagnosing students' misconceptions in algebra: Results from an experimental pilot study[END_REF]. The test is designed to identify the conceptual errors students make when working with a particular concept. The test results could inform the teacher who can then guide instruction aimed at alleviating these conceptual errors (ibid.:2009). It is possible that many teachers do not realise the essential connections that algebra has to numerous other mathematical content areas, and that how conceptual errors in one or more of these areas can hinder a students' progress, which will be discussed in more detail below [START_REF] Bush | Prerequisite algebra skills and associated misconceptions of middle grade students: A review[END_REF]. This test, in conjunction with an appropriate framework to interpret students' answers, could help inform instruction. The teaching and learning of algebra in Ireland aligns closely with Kieran's (2004) model, which outlines three activities that learners of school algebra must participate in: 1. Generational activities, 2. Transformational activities and 3. Global/Meta Level activities. It is noted that two aspects of algebra underlie all others, namely generality and abstraction (Department of Education and Skills, 2016 p. 26). These aspects have led to the definition of three types of algebraic activities that mathematics students in Ireland must engage in -representational, transformational and activities involving generalising and justifying. The test items have been designed to support the implementation of the new syllabus and in line with the approach to algebra being used in Ireland.

One potential criticism of the approach used in developing this test instrument is that it is a test of mathematics globally, given the broad range of content areas included. However, difficulties with algebra lie with both algebra itself and also with other areas of mathematics that students will have encountered, which are seen as prerequisites for the learning and understanding of algebra [START_REF] Bush | Prerequisite algebra skills and associated misconceptions of middle grade students: A review[END_REF]. Problems and conceptual errors in any one of the content areas can lead to problems in another area thus hindering a student's progress in algebra as a whole. Although these prerequisite areas support other areas of mathematics learning such as number and numerical operations, they are identified in the literature as core areas for the learning and understanding of algebra [START_REF] Bush | Prerequisite algebra skills and associated misconceptions of middle grade students: A review[END_REF]. The following section outlines the content areas identified throughout the literature as pertinent to success in algebra, on which test items have been based.

Prerequisite and algebra content areas

As stated the focus of this paper is the key content areas required for success in algebra. These content areas are outlined in Table 1 [START_REF] Bush | Prerequisite algebra skills and associated misconceptions of middle grade students: A review[END_REF][START_REF] Warren | Patterns that support early algebraic thinking in the elementary school[END_REF]. -Use the equivalence of fractions, decimals and percentages to compare proportions.

Decimals and Percentages

Number Systems:

-Calculate percentages -Use the equivalence of fractions, decimals and percentages to compare proportions. 1 refers to the strand numbers, for instance 3.2 refers to section 2 of strand 3. The diagnostic test has been developed for use with students in second year and therefore it was important to align the test items specifically with the content of the CIC. Once the CIC is complete teachers use their own discretion to introduce their topics (Department of Education and Skills, 2016). There is no prescribed structure for following the syllabus however, it is desirable that students will have completed their basic algebra skills including equation solving by the end of first year (Project Maths, n.d.-a).

Integers

Prerequisite content areas

Difficulties with algebra lie with both algebra itself and also with other areas of mathematics that students will have encountered, which are seen as prerequisites for the learning and understanding of algebra. Proportional reasoning is a key aspect of numeracy and it leads to relational thinking which is important in the development of algebraic skills, it is highly conceptual and a skill that develops gradually. Equally, fractions are an integral part of algebra and can be found as coefficients, constants and solutions to equations, the slope of a line, and, in general, proportions are written in fraction form in algebra [START_REF] Bush | Prerequisite algebra skills and associated misconceptions of middle grade students: A review[END_REF]. Knowledge of decimals, their value and placement on a number line, computation with decimals, and the ability to convert between decimals, fractions and percentages is also important for success in algebra [START_REF] Bush | Prerequisite algebra skills and associated misconceptions of middle grade students: A review[END_REF]. Studies have been conducted to identify what element of decimal and fraction understanding best indicate a students' performance in algebra. It has been found that the relational understanding of the bipartite format of a fraction and unidimensional magnitude, measured with the placement of decimals on a number line are the best predictors [START_REF] Dewolf | From rational numbers to algebra: Separable contributions of decimal magnitude and relational understanding of fractions[END_REF].

In learning about fractions, decimals and percentages, with the use of number lines and graphs, students at junior cycle are expected be able to compare and order numbers. This provides students with the skills and knowledge to apply the rules correctly when working with variables. It also enables a student to assess if a solution to an equation or inequality is reasonable. However, if students do not understand a fraction, decimal or percent, they are unable to extend their understanding to which is greater than or less than or equivalent [START_REF] Bush | Prerequisite algebra skills and associated misconceptions of middle grade students: A review[END_REF].

Furthermore, a solid understanding and procedural fluency with integers is required for success in algebra. Misconceptions about negative integers can impede progression, where, for example, a student may fail to accept a negative number as a solution to an equation. Research has suggested that the number line and graphs of functions can be used to help correct students' misunderstandings and conceptual errors with integers [START_REF] Bush | Prerequisite algebra skills and associated misconceptions of middle grade students: A review[END_REF]. Equally, an understanding of exponents is required for both the transformational skills, in dealing with expressions, and the generational and global/meta level skills, where knowledge of the shape of functions are required [START_REF] Bush | Prerequisite algebra skills and associated misconceptions of middle grade students: A review[END_REF]. Moreover, to succeed in the transformational rules of algebra it is essential to understand the order of operations. Some students believe that order of operations do not matter, that the same answer will result regardless. Others believe that the context of the problem determines the order of operations and in the absence of context operations should be performed from left to right. Research suggests that students should learn the hierarchy of operations more naturally by attending to more complicated operations first [START_REF] Bush | Prerequisite algebra skills and associated misconceptions of middle grade students: A review[END_REF].

Finding equivalent expressions is frequently required in algebra, and this manipulation requires an underlying sense of the properties of numbers. Allowing students to investigate the properties of numbers will assist in learning, retaining knowledge and developing relational understanding, which in turn will create a strong foundation for algebra [START_REF] Bush | Prerequisite algebra skills and associated misconceptions of middle grade students: A review[END_REF]. Numerous studies have focused on development of the concept of the equal sign in the early stages of learning algebra [START_REF] Bush | Prerequisite algebra skills and associated misconceptions of middle grade students: A review[END_REF]. Students often misinterpret the meaning of the sign viewing it as an operational sign.

Those who interpret the equal sign correctly and see it as a relational symbol have more flexibility when working with equations.

Algebra content areas [START_REF] Kieran | The learning and teaching of school algebra[END_REF] asserts that many misconceptions and common errors in algebra are generally rooted in the meaning of symbol or the letters used in algebra. Much research has been conducted into students' difficulties in working with algebraic variables and the misconceptions student's hold. These misconceptions include viewing variables as labels, the belief that the value of a variable has something to do with its position in the alphabet, and the belief that a variable is just a missing value rather than something which has varying values. These difficulties are then compounded when a student attempts to create and manipulate an algebraic expression [START_REF] Bush | Prerequisite algebra skills and associated misconceptions of middle grade students: A review[END_REF].

The underlying misconceptions and difficulties students hold in relation to variables, expressions and indeed all the prerequisite content areas can then lead to difficulties in solving algebraic equations. The ability to solve equations is reliant on both procedural and conceptual understanding. Conceptual understanding is strongly related to student's equation solving performance, as without it students learn by rote a series of transformational rules for dealing with equations. A solid understanding of how to use variables to write algebraic expressions, form subsequent equations and solve when necessary is the essence of success in algebra at junior cycle level [START_REF] Bush | Prerequisite algebra skills and associated misconceptions of middle grade students: A review[END_REF].

A function is defined as a correspondence between two sets [START_REF] Kieran | The learning and teaching of school algebra[END_REF], and there are two general approaches to teaching and learning functional relationships mentioned in the literature; a correspondence approach and a covariation approach [START_REF] Ayalon | Functions represented as linear sequential data: relationships between presentation and student responses[END_REF]. The correspondence approach deals with an input-output model, whereby an output value y is calculated for a given input value x, often listed in a table of values or as couples. This approach allows for determining the rule which generates the y-value from the x-value and is in line with the approach to teaching functions in Ireland. The concept of a function is not simple when you consider that at least three representations are used to convey the notion of a function; a table, a graph and an equation.

True procedural fluency and competency in working with functions is obtained when one can move between the different representations of a function with ease and this aligns with the multirepresentations approach advocated in the Irish syllabus [START_REF] Bush | Prerequisite algebra skills and associated misconceptions of middle grade students: A review[END_REF]Project Maths, n.d.-a) Finally, algebra can be seen as the language used to describe patterns and relationships for the ultimate goal of problem solving and as a systematic way of expressing generality (Project Maths, n.d.-a).

Students at junior cycle learn to identify the relationship which lies between the pattern and its position is a functional relationship meaning an expression or formula must be created using variables. In doing this a context for the use of variables is set for Irish students, assisting their understanding of a variable as a varying quantity rather than a specific unknown, laying down the foundation for understanding expressions and solving equations in what is known overall as a functions based approach to algebra (Project Maths, n.d.-a).

The diagnostic test

Test items were taken from previous relevant studies pertinent to measuring ability in the core content areas required for algebra outlined in Table 1. The diagnostic test currently contains twenty one questions summarized in Table 2 where the source of each test question is detailed. Students have space for workings and are then asked to circle the correct answer from the following options; A. from the original question include changing the word "integer" to "a whole number" and offering the "Hint", to ensure the question is more in line with the age profile of those being tested as informed by the pilot of this test and feedback from teachers. The above question was included as relational understanding of a fraction was an element identified in the DeWolf et al. ( 2015) study to predict performance in algebra. All test items have been developed with such a theoretical underpinning that is using multiple choice responses based on possible conceptual errors. In addition language was adjusted where necessary to make the test items more accessible for fourteen year old students.

Content

Conclusion

There are fresh concerns in relation to student attainment in mathematics in Ireland, specifically algebra and for progression to third level education [START_REF] Treacy | Trends in basic mathematical competencies of beginning undergraduates in Ireland, 2003-2013[END_REF]. There is a clear need to intervene early in the effort to address the issues students are facing with learning and understanding algebra. The overall aim in using this test is to identify conceptual errors that students make in both algebra and the prerequisite content areas required for success in algebra, therefore assisting to identify possible root causes of the students' errors, and as a result, through appropriate intervention improve students' knowledge of algebra and therefore general mathematical ability.

Ultimately, this will be a tool for teachers to use in the classroom allowing them to make informed decisions and to plan appropriate interventions [START_REF] Russell | Diagnosing students' misconceptions in algebra: Results from an experimental pilot study[END_REF].

  Systems: -consolidate their understanding of the relationship between ratio and proportion.4.4 Examining algebraic relationships:proportional relationshipsFractions3.1 Number Systems: -Investigate models to think about operation on fractions.

Figure 1 :

 1 Figure 1: Question 5 on the diagnostic test based on relational fraction knowledge

1𝑛

  gets very close to 1, B.

  3.1 Number Systems:-Investigate models, such as the number line, to illustrate the operations on integers

	Exponents	3.2 Indices
	Order of	3.1 Number Systems: -Appreciate the order of operations, including use of
	operations	brackets
	Properties of	3.1 Number Systems: -Investigate the properties of arithmetic and the
	numbers	relationships between them.
	Patterns	4.1 Generating arithmetic expressions from repeating patterns: -use tables and
		diagrams to represent a repeating-pattern situation -generalise and explain
		patterns and relationships in words and numbers -write arithmetic expressions
		for particular terms in a sequence.

Compare and order numbers

3.1 Number Systems:

-Use the number line to order natural numbers, integers and rational numbers. -Use the equivalence of fractions, decimals and percentages to compare proportions.

Equality

3.1 Number Systems: -Consolidate the idea that equality is a relationship in which two mathematical expressions hold the same value. Variables 4.6 Expressions: -Using letters to represent quantities that are variable. Algebraic expressions 4.6 Expressions: -Arithmetic operations on expressions. -Transformational activities Algebraic equations 4.7 Equations and inequalities: -Selecting and using suitable strategies for finding solutions to equations and inequalities. Functions 4.2 Representing situations with table diagrams and graphs: -use tables, diagrams and graphs as a tool for analysing relationspresent and interpret solutions, explaining and justifying methods, inferences and reasoning.

Table 1 : Content Areas for Diagnostic Test and alignment with the Junior Cycle syllabus

 1 Related content domains from the Irish syllabus were used as a framework to align the prerequisite and algebra content areas identified in the literature. The Number Strand (3) of the syllabus builds on primary school learning and facilitates the transition from arithmetic to Algebra (Strand 4). The Common Introductory Course (CIC) is the minimum course to be covered by all students at the beginning of the junior cycle, elements from the CIC are in italics within Table1. The numbering within Table

Areas Test Question Number and Source

	Ratio and proportion	1. Number Line/Decimal Magnitude from DeWolf et al. (2015)
		7. Proportional reasoning from Hilton, Hilton, Dole, and Goos (2013)
	Fractions	2. 4. 5. Fraction Knowledge from DeWolf et al. (2015)
		3. Fraction Knowledge multiplication (Bush & Karp, 2013)
	Decimals and	1. Number Line/Decimal Magnitude from DeWolf et al. (2015)
	Percentages	11. Comparing and Ordering Numbers, Project Maths (n.d.-b)
	Integers	15. Integers and equations adapted from Vlassis (2008)
	Exponents	6. and 8. adapted from discussion in Mok (2010)
	Order of operations	9. Order of operations adapted from Linchevski and Livneh (1999)
	Properties of numbers 10. Distributive property, adapted from discussion in Mok (2010)
	Comparing and	11. Comparing and Ordering Numbers, Project Maths (n.d.-b)
	ordering numbers
	Equality	12. 13. adapted from Stephens, Knuth, Blanton, Isler, Gardiner, and
		Marum (2013)
	Variables	14. Variable as label adapted from Küchemann (1981)
	Algebraic expressions 16. adapted from Hodgen, Kuchemann, Brown, and Coe (2009)
		17. Simplifying expressions based on errors discussed in Kieran (1992)
	Algebraic equations	20. adapted from Clement, Lochhead, and Monk (1981)
		18. 19. Next step of solution adapted from Chung and Delacruz (2014)
		21.3 Forming equations adapted from Ayalon et al. (2015)
	Patterns	21 Interpreting from a geometric pattern from Ayalon et al. (2015)

Table 2 : Summary of content and source of items on the diagnostic test

 2 An example of a test item, which assesses relational fraction knowledge, together with understanding of a variable and algebraic expression is taken and adapted from[START_REF] Dewolf | From rational numbers to algebra: Separable contributions of decimal magnitude and relational understanding of fractions[END_REF] as follows;n is a whole number greater than 0. If 𝑛 continues to get bigger in value, please circle one of the following options A, B or C in the answer box for what happens to

Proceedings of CERME10


Chris Winstead
email: chris.winstead@usu.edu

Tasnuva Tithi
email: tasnuvatithi@aggiemail.usu.edu

Emmanuel Boutillon
email: emmanuel.boutillon@univ-ubs.fr

Fakhreddine Ghaffari
email: fakhreddine.ghaffari@ensea.fr

Recent Advances on Stochastic and Noise Enhanced Methods in Error Correction Decoders

This paper offers a review of recent developments in non-deterministic error correction decoding methods, which can be described in two broad classes. The first class uses stochastic computation to emulate the arithmetic operations of conventional decoding algorithms. The second class achieves noise enhancement by randomly perturbing the calculations of a standard decoder. Stochastic decoders inherit analysis techniques from the conventional algorithms they emulate, but the noiseenhanced algorithms are newer, more difficult to explain, and not yet fully understood. We describe a Markov chain analysis technique to both explain and optimize noise enhancement in these algorithms. Circuit implementation is also discussed, including both conventional hardware architectures and circuits based on memristor threshold logic, where memristor non-determinism can be exploited for noise enhancement.

I. INTRODUCTION

Non-deterministic algorithms are well known in a variety of fields (optimization, imaging, neuromorphic computing, etc), in which the performance or efficiency is enhanced beyond what could be obtained by a deterministic method. For many years, there has been a corresponding interest in reliable or faulttolerant computing with non-deterministic physical components. In this paper we survey a family of non-deterministic error correction algorithms and circuit implementations that merge these topics, providing a useful framework for noise enhancement at the physical layer of data communication, memory and storage. Most of the recent advances in this area focus on Low Density Parity Check (LDPC) codes, a well known class of error correction algorithms that are now included in numerous data communication standards. The survey will therefore be limited to the domain of LDPC codes, though the concepts are by no means restricted to that class of codes.

LDPC codes are used to encode data by inserting redundant parity bits in anticipation of corruption due to noisy communication channels or faulty memory cells. Error correction is usually done using Bayesian Belief Propagation (BP) or related algorithms. BP-based algorithms perform exceptionally well in that they achieve very low error rates after decoding retrieved data. An alternative class of Bit-Flipping (BF) algorithms is also widely studied. BF decoders are much simpler to implement in hardware, but achieve comparatively poor error rates. During the past two decades, a great deal of research has been invested into designing algorithms that optimize the tradeoff between BP's excellent performance and BF's simplicity.

A. Review of stochastic decoding algorithms

LPDC decoders belong to a class of message-passing algorithms defined on factor graphs. A message passing decoder computes local probability or likelihood metrics at each node. Local calculations are iterated until all parity-check constraints are satisfied, indicating global convergence.

A typical BP-based decoder must perform a number of integer additions, lookup table operations, or comparisons to obtain a metric for every edge in the graph. In order to reduce the hardware cost of these decoders, beginning in 2003 researchers considered stochastic computing methods to emulate the required arithmetic [START_REF] Gaudet | Iterative decoding using stochastic computation[END_REF]. The basic concept is to compute probability metrics by filtering pseudorandom bit sequences that model a Bernoulli process. A probability value is directly encoded in the frequency of 1's appearing in the sequence. The hardware is often simpler than BPbased decoders: adders are replaced by Muller C-element gates (similar in function to J/K flip-flops). An example C-element implementation is shown in Fig. 1. In the figure, the processor receives two input Bernoulli streams, A and B, with statistics p A and p B , respectively. It is well known that the output is a sequence with statistic

p Q = p A p B p A p B + (1 -p A ) (1 -p B )
which corresponds to the BP symbol node operation.

For an acyclic graph, a stochastic decoder has performance equivalent to BP. Most code graphs are cyclic, and in this case the stochastic algorithm becomes trapped in deterministic patterns, resulting in uncorrectable error patterns along the cyclic subgraphs. To resolve this problem, the variable node implementation (Fig. 2) is modified to have non-deterministic behavior [START_REF] Winstead | Stochastic iterative decoders[END_REF]. The best performing modification is the family of "edge memory" methods, which restore performance close to BP [START_REF] Tehrani | Stochastic decoding of LDPC codes[END_REF]. An edge memory is a module that samples the statistics of an input bit stream, and emits a separate, independent stream with the same statistics. If an edge memory is inserted in every cycle within the decoder, then deterministic loops are eliminated while preserving the BP-based probability calculations.

B. Noise enhanced decoders

Arguably the first observation of non-deterministic enhancement was in 2001, in the context of analog decoders where device mismatch was observed to enhance performance in some cases [START_REF] Lustenberger | On mismatch errors in analog-VLSI error correcting decoders[END_REF]. This phenomenon was explained by the suboptimality of BP on cyclic code graphs, which opened a possibility for accidental improvement in randomly altered designs. A more systematic approach to noise enhancement appeared with probabilistic bit flipping (PBF) in 2005, where Gallager-style bit-flipping algorithm was modified to flip bits only with probability p (i.e. the algorithm's normal bit flips are suppressed with probability 1 -p) [START_REF] Miladinovic | Improved bit-flipping decoding of low-density parity-check codes[END_REF]. In 2009, noise enhancement was observed for stochastic decoders implemented with a mixture of permanent defects and transient errors [START_REF] Winstead | A probabilistic LDPC-coded fault compensation technique for reliable nanoscale computing[END_REF], where noise-induced upsets were found beneficial for compensating permanent faults.

In 2012, dithered belief propagation was described, marking the first explicit construction of a noise enhanced BP decoder where decoding operations are deliberately perturbed by artificial noise [START_REF] Leduc-Primeau | Dithered belief propagation decoding[END_REF]. This was followed soon after by four developments in 2014. First, noise enhancement was observed for the BP-based Min-Sun (MS) algorithm on the binary symmetric channel (BSC) [START_REF] Ngassa | Unconventional behavior of the noisy min-sum decoder over the binary symmetric channel[END_REF]. In the same year, Noisy Gradient Descent Bit Flipping (NGDBF) was described for the Gaussian channel [START_REF] Sundararajan | Noisy gradient descent bit-flip decoding for LDPC codes[END_REF] and the related Probabilistic Gradient Descent Bit Flipping (PGDBF) algorithm for the BSC [START_REF] Al Rasheed | Fault-tolerant probabilistic gradient-descent bit flipping decoder[END_REF]. These works led to a number of algorithm and hardware improvements for NGDBF [START_REF] Sundararajan | ASIC design of a noisy gradient descent bit flip decoder for 10GBASE-T ethernet standard[END_REF]- [START_REF] Dai | Noisy gradient descent bitflipping decoder based on adjustment factor for LDPC codes[END_REF] and PGDBF [START_REF] Le | Hardware optimization of the perturbation for probabilistic gradient descent bit flipping decoders[END_REF]- [START_REF] Le | Variable-node-shift based architecture for probabilistic gradient descent bit flipping on QC-LDPC codes[END_REF], along with potential specialized applications in solid-state storage [START_REF] Ivaniš | Error errore eicitur: A stochastic resonance paradigm for reliable storage of information on unreliable media[END_REF]. A related approach, known as the Improved Differential Binary (IDB) decoder, was also proposed in 2014 [START_REF] Cushon | Highthroughput energy-efficient LDPC decoders using differential binary message passing[END_REF]. The IDB decoder uses a combination of feedback (similar to the stochastic TFM) and perturbed threshold operations.

The GDBF algorithm is based on sum-and-threshold operations, as shown in Fig. 2. First, a hard decision syndrome vector is computed by the parity-check nodes. Then, for each variable node, an "energy function" is computed (also called a "reliability" or "flipping" function) by the expression

E (GDBF) k = x k y k + j∈M k s j , (1)
where y k is the channel information, x k ∈ {+1, -1} is the bipolar hypothesis decision, M k is the parity-check

Σ θ + q + y s 1 . . . s dc T Q x clk + - initialize to sign(y)
Fig. 2: Typical variable node implementation for the NGDBF algorithm based on a toggle flip-flop. Inputs are channel information y, threshold θ, random perturbation q, and adjacent parity-check messages s j .

neighborhood of the k th variable node, and the s j are bipolar syndrome values (parity is satisfied when s j = +1 and violated when s j = -1). In each iteration, the decision x k is flipped if

E k < θ,
where θ is a threshold that may vary from iteration to iteration or, in some algorithms, from node to node.

In the NGDBF algorithm, the threshold operation is perturbed by injected noise q:

E (NGDBF) k = x k y k + j∈M k s j + q, (2)
In the PGDBF algorithm, the sum-and-threshold operation is carried out using the GDBF rule (1), and if E k < θ then the bit is flipped only with probability p flip . In the NGDBF and PGDBF algorithms, some kind of random number generator (RNG) is needed to generate the non-deterministic behavior. For NGDBF, Gaussian perturbations have generally been used, but other distributions have also been used with a small performance penalty [START_REF] Sundararajan | Noisy gradient descent bit-flip decoding for LDPC codes[END_REF]. There are now numerous variations on the NGDBF and PGDBF algorithms, which differ mainly in the method and scheduling of how θ is determined.

Besides bit-flipping and BP-based decoders, other types of noise-enhanced decoders have been reported, such as a recent stochastic BCH decoder [START_REF] Han | A hardware efficient soft-decision BCH decoder for WBAN systems using stochastic computing[END_REF] and a stochastic Chase algorithm [START_REF] Leroux | Stochastic Chase decoding of Reed-Solomon codes[END_REF]. These are often described as stochastic decoders, however they do not utilize the hardware techniques traditionally labeled as "stochastic computing" and are arguably more similar to noise-enhanced decoders than to the BP-based stochastic decoders described in Sec. I-A, since they perform a discrete codeword search aided by non-determinism. Research on noisy BP-based decoders has also continued to advance, as with the recent Noise-Against-Noise Decoder (NAND) algorithm [START_REF] Cochachin | Density evolution thresholds for noise-against-noise min-sum decoders[END_REF].

II. THEORY

The development of noise-enhanced decoders was experimentally driven, and significant performance enhancing effects were discovered and optimized empirically prior to developing a sound theoretical footing. The empirical approach nevertheless realized a high degree of performance gain. One example is shown in Fig. 3, which compares performance of stochastic and NGDBF algorithms with ideal BP decoding on the AWGN channel. Compared to noiseless GDBF, the leap in performance due to solely to noise is particularly striking. A second example is shown in Fig. 4, which shows performance of a "decoder-dynamic shift" (DDS) PGDBF algorithm [START_REF] Declercq | Noise-aided gradient descent bit-flipping decoders approaching maximum likelihood decoding[END_REF] approaching maximum-likelihood performance on the BSC when a large iteration limit is allowed. The noise gains depicted in Figures 3 and4 were discovered essentially by accident, and theoretical explanations came later. To date, the most productive explanation is that noise tends to disrupt trapping sets in the GDBF decoder. Trapping set analysis aids in parametric optimization for noisy decoders, but does not support ensemble optimizations like the ones obtained with density evolution for BP-based decoders. LDPC codes are represented by compact graphs with many cycles, which break the formal correctness of BP and related algorithms. Certain error patterns can be difficult or impossible to correct on cyclic subgraphs known as trapping sets. In the case of BP-based algorithms, trapping sets are well known as the cause of error floors. In the domain of bit-flipping algorithms, trapping sets have proven useful for understanding and optimizing noise enhancement effects.

A Markov process approach to trapping sets was previously described [START_REF] Declercq | Noise-aided gradient descent bit-flipping decoders approaching maximum likelihood decoding[END_REF], and is here illustrated by a simple example. We consider a trapping set involving m variable nodes. Then there are 2 m possible binary states for the variable nodes involved in the trapping set. Only one of these states is correct, and without loss of generality we assume it is state 0. Since the decoding algorithm is non-deterministic, in each iteration it may undergo a transition from its present state s i to any other allowed state s j with probability p ij , hence the decoding iterations constitute a homogeneous Markov chain characterized by a square transition matrix Γ of dimension 2 m with entries p ij .

For an initial erroneous state s 0 , the probability that the error is corrected after L iterations is the zeroth element of s 0 Γ L . Since Γ and s 0 depend on the received channel information, we use a Markov Chain Monte Carlo procedure to evaluate the trapping set's contribution to the error probability over many frames. This procedure reveals that noise is essential for BF algorithms to correct trapping set errors. Furthermore, since Γ is dependent on various algorithms parameters (e.g. the noise statistics and distribution, weighting or clipping of channel samples, etc), these parameters can be adjusted and optimized at design time [START_REF] Trung | New direction on low complexity implementation of probabilistic gradient descent bit-flipping decoder[END_REF].

III. IMPLEMENTATION

A. Conventional ASIC and FPGA architectures ASIC implementations of stochastic and noise enhanced BF decoders usually are fully parallel architectures that precisely map the code's factor graph topology. The circuit implementations have few degrees of freedom for the variable and check nodes, hence most design effort tends to focus on generating and distributing random numbers or on the design of efficient edge memories. Recent stochastic decoders proved competitive with BP-based decoders in both performance and silicon area [START_REF] Tehrani | Majority-based tracking forecast memories for stochastic LDPC decoding[END_REF]. More recent BF implementations require 2× to 3× less area than stochastic decoders, with similar or better decoding performance. The area reduction is explained by the data flow architectures shown in Fig. 5. In every iteration, stochastic decoders must compute two messages for every edge in the factor graph, whereas the BF decoders need only compute one message per node.

The normalized power consumption of BF decoders, reported in Joules per bit, is the lowest among decoders implemented in comparable technology. Due to their non-deterministic convergence, stochastic and noisy BF decoders cannot guarantee the same latency for all frames, and may require buffering frames to accommodate occasional delays. The average throughput is nevertheless high enough for major data communication standards, such as IEEE 802.3an 10GBase-T [START_REF] Sundararajan | ASIC design of a noisy gradient descent bit flip decoder for 10GBASE-T ethernet standard[END_REF], [START_REF] Tehrani | Majority-based tracking forecast memories for stochastic LDPC decoding[END_REF].

Non-deterministic BF decoders have been reported to perform well with a variety of non-ideal random number sources. Where the earliest designs typically employed linear feedback shift registers (LFSRs) to produce random numbers, it would be prohibitive to include an LFSR in every variable node. This motivated random sample reuse, where a large shift register is used to distribute samples from a single RNG [START_REF] Sundararajan | Noisy gradient descent bit-flip decoding for LDPC codes[END_REF]. Simpler approaches use no RNG at all, as random samples can be pre-generated as initialization values for a circular shift register [START_REF] Sundararajan | ASIC design of a noisy gradient descent bit flip decoder for 10GBASE-T ethernet standard[END_REF]. Another LFSR-free approach uses an "intrinsic valued random generator" to extract random numbers from switching activity within the decoder [START_REF] Le | Efficient hardware implementation of probabilistic gradient descent bit-flipping[END_REF].

For quasi-cyclic LDPC (QC-LDPC) codes, the recently described Variable Node Shift Architecture (VNSA) implements PGDBF without requiring any RNG [START_REF] Le | Variable-node-shift based architecture for probabilistic gradient descent bit flipping on QC-LDPC codes[END_REF]. In the VNSA, a unique pre-generated noise sample is hard-coded into each variable node. The code's structure permits cyclically shifting the variable node states after each iteration. With each iteration, the variable nodes' logical positions are shifted to new physical locations, where they obtain an updated noise sample from the local hardware instantiation. The VNSA decoder has hardware area almost as low as GDBF, making it one of the most area efficient decoder architectures.

B. Unconventional emerging devices: memristors

In the development of stochastic and noise-enhanced decoders, two central points emerged: first, deterministic logic is fatal for stochastic decoders with cycles; second, nondeterminism is essential for performance gain in noise enhanced decoders. In hardware implementation, non-determinism is usually provided using RNG circuits that incur substantial overhead in area and power. There is now an interest in exploiting nanoscale "native non-determinism" as a computational resource to mitigate the hardware cost of RNGs.

Recent research points to resistance-switching devices, commonly identified as a subclass of memristors, as having nondeterministic switching statistics that can be modeled with some precision, and can be adjusted by manipulating pulse timing or amplitude in memristive logic circuits [START_REF] Al-Shedivat | Memristors empower spiking neurons with stochasticity[END_REF]. A memristor of this type can be induced to switch between high-resistance and low-resistance states by applying forward and reverse voltages (respectively) in excess of the device's threshold voltage. When a lesser sub-threshold voltage is applied, the device's switching is a stochastic process. This phenomenon already attracted interest from the neuromorphic circuits community, where stochastic switching provides a model for neural networks based on spike timing dependent plasticity (STDP) [START_REF] Al-Shedivat | Memristors empower spiking neurons with stochasticity[END_REF], [START_REF] Naous | Stochasticity modeling in memristors[END_REF].

Memristor circuits were previously considered for LDPC decoders [START_REF] Poikonen | Memristive circuits for LDPC decoding[END_REF], and for stochastic computation [START_REF] Knag | A native stochastic computing architecture enabled by memristors[END_REF]. A possible memristor-based randomness source is shown in Fig. 6. The memristor is initialized in a high-resistance state by applying a reverse pulse V rst which exceeds the memristor's switching threshold. A train of sub-threshold pulses are then applied at V p . A sub-threshold pulse can cause the memristor to switch to a low-resistance state with probabiilty p sw . The switching statistics are a function of the pulse amplitude and width [START_REF] Al-Shedivat | Memristors empower spiking neurons with stochasticity[END_REF]. The memristor's resistance state is detected by monitoring the voltage in the resistor divider circuit formed by R. Each time the memristor switches to a low-resistance state, a reset cycle is required to restore the memristor to its high-resistance condition. The digital output Q is a true random Bernoulli sequence. Memristor-based random generators therefore have the potential to produce random perturbations with superior statistical features (i.e. true random instead of pseudo-random sequences), with less circuit area than traditional RNGs.

IV. CONCLUSION

This paper gave an abbreviated review of recent progress in stochastic and noise-enhanced decoding, emphasizing advances during the past five years. Progress in this topic appears to be accelerating and there are now applications for commercial standards. Better theoretical tools are still needed to understand convergence behavior and optimize throughput, to establish performance bounds, and to determine optimal code constructions for use with these algorithms. With the introduction of memristor circuits, new territory is now opened for continued research and discovery bridging the physical and algorithmic foundations of communication and memory.

Fig. 1 :

 1 Fig. 1: Stochastic variable node implementation based on a Muller C-element with J/K flip-flop.

Fig. 3 :

 3 Fig.3: Example of noise-enhanced performance for stochastic, IDB and NGDBF algorithms, applied to the IEEE 802.3an 10GBase-T standard. NGDBF allows a maximum of 600 iterations to decode a frame.

Fig. 4 :

 4 Fig.4: Example of ML-approaching performance for the PGDBF algorithm on the Tanner code with N = 155. For each algorihtm, the maximum allowed iterations per frame are indicated in parentheses.

  t p e r m e s s a g e → parity messages (1 per node) ← 1 bit per message Σ (b) True bit-flipping decoder.

Fig. 5 :

 5 Fig. 5: Comparison of iterative data flow in stochastic vs BF architectures.

Fig. 6 :

 6 Fig. 6: A memristor-based randomness source.

ACKNOWLEDGEMENT

This work was partly supported by the US National Science Foundation under award number 1410000, by the State of Utah USTAR initiative under the UTAG program, and by the french ANR under grant number ANR-15-CE25-0006-01 (NAND project).


Erwann Legrand
email: erwann.legrand@sb-roscoff.fr

Sophie Martin

Cédric Leroux

Pascal Riera

Effect of

chin Psammechinus miliaris were maintained for 1 month at 17°C (mean summer temperature in the Bay of Brest) and at 20°C (maximum summer temperature) and fed with 13 C-and 15 N-labeled Solieria chordalis. The results showed that the urchins' 13 C uptake was 0.30 µg 13 C g dry weight (DW) -1 at 17°C and 0.14 µg 13 C g DW -1 at 20°C at the end of the experiment. The lower uptake at the higher temperature may be attributed to a decrease in metabolic activity at 20°C, involving lower feeding and/or respiration rates. Conversely, no significant effect of temperature was detected on 15 N uptake. At the end of the experiment, the urchins' 15 N uptake was 0.04 µg 15 N g DW -1 at 17°C and 0.03 µg 15 N g DW -1 at 20°C. This suggests that temperature may affect carbon and nitrogen trophic fluxes differently. The use of dual isotope labeling offers interesting prospects and needs to be further extended in order to better understand trophic interactions in marine communities and the consequences of current environmental changes, such as global warming.

K E Y W O R D S

carbon, macroalga, nitrogen, stable isotope labeling, temperature, trophic interaction, uptake, urchin In the Bay of Brest, the alga S. chordalis is abundant [START_REF] Hily | Structure of subtidal algal assemblages on soft-bottom sediments: fauna/flora interactions and role of disturbances in the Bay of Brest, France[END_REF] and reaches the highest biomass in the summer [START_REF] Guillou | Can low sea urchin densities control macro-epiphytic biomass in a north-east Atlantic maerl bed ecosystem (Bay of Brest, Brittany, France)?[END_REF]. Therefore, trophic transfers were investigated at 17°C (mean summer temperature, recorded by Service d'Observation en Milieu LITtoral, SOMLIT) and at 20°C (maximum summer temperature, [START_REF] Martin | Community metabolism in temperate maerl beds. I. Carbon and carbonate fluxes[END_REF].

| MATERIAL AND ME THODS

| Species collection

| Experimental design

In May 2016, 36 Psammechinus miliaris individuals of medium size (26 ± 3 mm) were randomly assigned to 10 15-l aquaria (Supporting Information Fig. S1). In all aquaria, the temperature was gradually increased by 0.5°C per day over 4 days to reach 17°C, corresponding to the mean summer temperature in the Bay of Brest (SOMLIT).

Then, in five aquaria, the temperature was increased again for 6 days to reach 20°C, the mean maximum summer temperature in the Bay of Brest [START_REF] Martin | Community metabolism in temperate maerl beds. I. Carbon and carbonate fluxes[END_REF]. The temperature was controlled in two 100-l tanks, continuously supplied with filtered (5 µm) open natural seawater, with a high water flow rate of 150 L/hr per tank. The temperature was maintained by an offline feedback system (IKS Aquastar, Karlsbad, Germany) that activated or stopped heaters, controlling the temperature in the tanks. Each 100-L tank provided seawater to five aquaria. During the experiment, the temperature was at 17.2 ± 0.3 and at 19.9 ± 0.2°C, according to the treatments (17 and 20°C). For both temperature treatments, urchins were starved 10 days before the feeding experiment to empty their gut. During the experiment, the irradiance was 90-100 µmol photons m -2 s -1 , corresponding to the mean in situ summer irradiance at 5 m depth on maerl beds in the Bay of Brest [START_REF] Martin | Primary production, respiration and calcification of the temperate free-living coralline alga Lithothamnion corallioides[END_REF].

| Seaweed labeling

Thalli of Solieria chordalis were cleaned of epiphytes, rinsed with filtered seawater (on 0.7 µm GF/F glass microfiber filters) and placed in a 9-L closed-circuit aquarium. Five liters of filtered (0.7 µm) seawater was added in the aquarium. A solution of 13 C-HCO 3 -at 1 g/L was prepared from NaH 13 CO 3 (99 atom% 13 C) and 1 L of filtered 0.7 µm GF/F seawater. Similarly, a solution of 15 N-NH 4 at 0.25 g/L was prepared by diluting 15 NH 4 Cl (99 atom% 15 N) in filtered seawater. The seaweeds were fertilized with the solutions of NaH 13 CO 3 and 15 NH 4 Cl for a period of 7 days. Each day, 10 ml of a solution of phosphate (PO 4 3-

) at 1 g/L was also added to ensure that P was not limiting. A circulation pump ensured nutrient homogeneity in the aquarium. At the end of the labeling period, some of the thalli were rinsed with filtered seawater and immediately frozen at -20°C until isotope measurements. In order to provide sufficient food for urchins for the duration of the experiment, a second stock of S. chordalis was labeled after 8 days, following the same protocol.

| Feeding experiment

For each temperature treatment, three urchins were collected at day 0 (T0) to measure the initial isotopic signature. Labeled thalli of Solieria chordalis from the same labeled stock were provided every 2 days to urchins of eight aquaria (four per condition) at an amount of 1 g of fresh weight per urchin. In one control aquarium per temperature treatment, urchins were fed with non-labeled thalli at the same amount (1 g per urchin). After 6 days, one urchin was collected per aquarium. A second urchin was collected after 15 days and the last urchin after 30 days (T30). Just before collection, urchins were starved for 48 hr to allow the assimilation of ingested algae and to allow evacuation of their digestive content. All samples were immediately frozen at -20°C pending isotope measurement.

| Isotope analysis

For each individual, the digestive system was removed and samples were acidified under a magnifying glass to remove calcareous structures [START_REF] Kamp | Processing of 13 C-labelled phytoplankton in a fine-grained sandy-shelf sediment (North Sea): relative importance of different macrofauna species[END_REF]. Urchin soft tissues and algae were dried in an oven for 48 hr at 60°C, then ground to a fine powder.

Subsamples were weighed into tin cups for isotope analyses. Stable isotope ratios for carbon (R = 13 C/ 12 C) and nitrogen (R = 15 N/ 14 N)

were determined using a carbon, hydrogen and nitrogen (CHN) analyser (ThermoFinnigan 1112 Series) interfaced with a mass spectrometer (ThermoFinnigan MAT Deltaplus) via a Conflow III open split interface.

| Uptake of 13 C and 15 N by urchins

Carbon and nitrogen ratios were used to calculate δ-values (in ‰):

where X is 13 C or 15 N. Sample ratios (R sample ) were reported relative to international standards (R standard ) Vienna Pee Dee Belemnite (R VPDB = 0.0112372) for carbon and atmospheric N 2 (R air N2 = 0.0036765) for nitrogen. Incorporation of 13 C or 15 N labels in urchins was calculated as excess 13 C or 15 N, expressed in terms of specific uptake (i.e.,

Δδ 13 C = δ 13 C T30δ 13 C T0 and Δδ 15 N = δ 15 N T30δ 15 N T0 ) and total uptake (I), in accordance with previous studies [START_REF] Herman | Stable isotopes' as trophic tracers: combining field sampling and manipulative labelling of food resources for macrobenthos[END_REF]; [START_REF] Leroy | Importance of bacterivory and preferential selection toward diatoms in larvae of Crepidula fornicata (L.) assessed by a dual stable isotope ( 13 C, 15 N) labeling approach[END_REF]. I corresponds to the product of excess 13 C or 15 N (E) and carbon and nitrogen biomass (B), with I in µg 13 C or 15 N, B in µg C or N. E is calculated as the difference between the fraction 13 C or 15 N of the initial value (F T0 ) and the final value (F T30 ):

δX = [(R sample ∕R standard ) -1] × 1,000
with F = 13 C/( 13 C + 12 C) = R/(R + 1) for 13 C and F = 15 N/ ( 15 N + 14 N) = R/(R + 1) for 15 N. The carbon and nitrogen isotopic ratios (R) were obtained from the measured δ 13 C and δ 15 N, respectively, as:

with R VPDB the Vienna Pee Dee Belemnite-limestone standard, and, with R air N2 the atmospheric dinitrogen standard.

Total carbon and nitrogen transfer ( 12 C and 13 C; 14 N and 15 N) from algae to grazers was obtained from 13 C and 15 N incorporation, dividing I 13 C or I 15 N of urchins by F 13 C or F 15 N of the source.

| Data analysis

All data were expressed as the mean ± SE. Statistical analysis was completed using the statistical software R version 3.2.2 (R Core

| RE SULTS

After 8 days, stocks of Solieria chordalis thalli were labeled at a level of δ 13 C = 1152 ± 106‰, (n = 3) and 2131 ± 617‰, (n = 3) for stocks 1 and 2, respectively. These levels were significantly higher than non-labeled thalli (δ 13 C = -17.6 ± 0.4‰; Wilcoxon-Mann-Whitney test, p = .005). Similarly, labeled S. chordalis were significantly 15 Nenriched (δ 15 N = 5191 ± 701‰, n = 3 and 4604 ± 319‰, n = 3, for stocks 1 and 2, respectively) compared with non-labeled seaweeds (δ 15 N = 16.2 ± 0.3‰; Wilcoxon-Mann-Whitney test, p = .005).

At the beginning of the experiment, Psammechinus miliaris exhibited an initial δ 13 C level of -17.2 ± 0.7‰ (n = 3) and -18.4 ± 0.3‰

(n = 3) at 17 and 20°C, respectively. In urchins supplied with labeled S. chordalis, δ 13 C reached 941 ± 203‰ (n = 4) at 17°C and 623 ± 118‰ (n = 4) at 20°C after 30 days (Figure 1a). This enrichment was significant compared with the initial isotopic signatures (Wilcoxon-Mann-Whitney, p < .001). Throughout the experiment, higher δ 13 C values were observed at 17 than at 20°C, whereas no significant effect of time was observed (Table 1, Figure 1a).

Psammechinus miliaris initial δ 15 N values were 5.3 ± 0.8‰

(n = 3) and 4.4 ± 0.3‰ (n = 3) at 17 and 20°C, respectively. After 30 days, δ 15 N in urchins supplied with labeled S. chordalis reached 1405 ± 109‰ (n = 4) at 17°C and 1514 ± 335‰ (n = 4) at 20°C

(Figure 1b). The 15 N enrichment after 30 days was significant in comparison with initial isotopic values (Wilcoxon-Mann-Whitney, p < .001; Figure 1b). Temperature and time had no significant effect on urchins' δ 15 N (Table 1).

At 17°C, mean 13 C specific uptake in urchins supplied with labeled S. chordalis ranged from 0.22 µg 13 C (day 15) to 0.36 µg 13 C (day 6) and reached 0.30 µg 13 C after 30 days (Figure 2a). At 20°C, 13 C uptake ranged from 0.14 µg 13 C (day 30) to 0.19 µg 13 C (day 6). 13 C uptake was significantly higher at 17 than at 20°C (Table 2), whereas no significant effect of time was evidenced (Table 2). Psammechinus at 17°C (Figure 2b). At 20°C, 15 N uptake varied between 0.03 µg 15 N (day 30) and 0.05 µg 15 N (day 6). Temperature had no significant effect on 15 N specific uptake in P. miliaris, whereas 15 N uptake decreased significantly over time (Table 2).

E = F T30 -F T0 R = (δ 13 C∕1,000 + 1) × R VPDB R = (δ 15 N∕1,000 + 1) × R airN2 F I G U R E

| D ISCUSS I ON

In this experiment, the urchin Psammechinus miliaris consumed substantial amounts of the alga Solieria chordalis at 17 and 20°C. The gut transit time of P. miliaris varies according to the food source ingested [START_REF] Bedford | Macrofaunal involvement in the sublittoral decay of kelp debris: The sea urchin Psammechinus miliaris (Gmelin) (Echinodermata: Echinoidea)[END_REF] and may reach 42 hr in small individuals (Otero-Villanueva, 2004). The present feeding experiment was performed over 30 days, which is sufficient to measure carbon and nitrogen assimilation rates in P. miliaris.

At 17°C, an increase in δ 13 C and δ 15 N occurred during the first six days, corresponding to a significant 13 C and 15 N uptake. This may be a result of a high level of consumption of the labeled S. chordalis following the starving period that occurred before the experiments, as recently observed by [START_REF] Rubilar | Ingestion, absorption and assimilation efficiencies, and production in the sea urchin Arbacia dufresnii fed a formulated feed[END_REF]. In addition, several studies have reported a link between feeding and other metabolic processes in echinoids, such as respiration or excretion [START_REF] Brockington | Seasonality of respiration and ammonium excretion in the Antarctic echinoid Sterechinus neumayeri[END_REF][START_REF] Carr | Warming increases the top-down effects and metabolism of a subtidal herbivore[END_REF]. These physiological processes can influence the 13 C/ 12 C and 15 N/ 14 N ratios. Among the physiological processes that can influence the 13 C/ 12 C and 15 N/ 14 N ratios, the increase in respiration rates induced by higher levels of feeding may contribute to 13 C enrichment due to a preferential loss of 12 C [START_REF] Rau | Animal 13 C/ 12 C correlates with trophic level in pelagic food webs[END_REF], and higher 15 N enrichment may be due to an increase in 15 N depleted nitrogen excretion [START_REF] Minagawa | Stepwise enrichment of 15 N along food chains: Further evidence and the relation between δ 15 N and animal age[END_REF] in the form of urea [START_REF] Fe R E N C E S Basuyaux | Inorganic nitrogen and its effect on growth of the abalone Haliotis tuberculata Linnaeus and the sea urchin Paracentrotus lividus Lamarck[END_REF]. At 20°C, although the food source was identical, the increases in δ 13 C and δ 15 N of P. miliaris during the first 6 days were less pronounced as compared to those at 17°C. Hence, from the present study, temperature appears to play an important role in the trophic fluxes from algae to grazers, especially regarding carbon uptake. 13 C uptake was significantly higher at 17 than at 20°C throughout the experiment. At the end of the experiment, urchins supplied with labeled 13 C showed an assimilation of 10.1 and 4.9 ng 13 C day -1 at 17 and 20°C, respectively. The effect of temperature increase on the 13 C uptake of P. miliaris appeared to significantly alter the total carbon uptake, as estimated total carbon fluxes after 30 days were 0.29 µg C g dry weight (DW) -1 day -1 at 17°C and 0.14 µg C g DW -1 day -1 at 20°C. These differences in carbon uptake observed between the two temperatures may be attributed to the higher metabolic activity at 17 than at 20°C, involving higher feeding and respiration rates. The lower metabolism at 20°C contrasts with the findings of several studies on diverse marine taxa (e.g., bivalves, gastropods, echinoderms), which showed higher feeding Abasolo-Pacheco, Mazón-Suastegui, [START_REF] Cáceres-Puig | Effect of temperature on growth and survival of Crassostrea corteziensis spat during late-nursery culturing at the hatchery[END_REF][START_REF] Carr | Warming increases the top-down effects and metabolism of a subtidal herbivore[END_REF][START_REF] Gooding | Elevated water temperature and carbon dioxide concentration increase the growth of a keystone echinoderm[END_REF][START_REF] Noisette | Metabolic responses to temperature stress under elevated pCO 2 in Crepidula fornicata[END_REF]. However, a previous experiment on the high-latitude urchin

Strongylocentrotus droebachiensis evidenced a decrease in feeding rates above 15°C, which is close to the maximum summer temperature (16°C, [START_REF] Frey | Thermal and hydrodynamic environments mediate individual and aggregative feeding of a functionally important omnivore in reef communities[END_REF]. Similarly in the lobster Jasus edwardsii, lower feeding and respiratory rates were observed above the upper thermal limit at 24°C [START_REF] Thomas | The effect of temperature on survival, growth, feeding and metabolic activity of the southern rock lobster, Jasus edwardsii[END_REF].

Interestingly, 13 C and 15 N uptake responded differently during the experiment. Indeed, conversely to 13 C uptake, no effect of temperature increase was observed on 15 N uptake in P. miliaris. At the end of this feeding experiment, 15 N incorporation was 1.2 ng 15 N day -1 at 17°C and 0.8 ng 15 N day -1 at 20°C, which corresponded to estimated total nitrogen fluxes of 17.8 and 13.2 ng N g DW -1 day -1 , respectively. This suggests an effect of temperature increase on carbon and nitrogen stoichiometry in P. miliaris. Although heterotrophs are considered as more homeostatic than autotrophs [START_REF] Persson | To be or not to be what you eat: regulation of stoichiometric homeostasis among autotrophs and heterotrophs[END_REF], several biological and environmental factors have been evidenced to influence stoichiometric homeostasis. Some studies have showen that changes in elemental composition of food sources may induce imbalance in consumer elemental homeostasis [START_REF] Malzahn | Effects of poor food quality on copepod growth are dose dependent and non-reversible[END_REF][START_REF] Prado | Moderate stoichiometric homeostasis in the sea urchin Lytechinus variegatus: effects of diet and growth on C:N: P ratios[END_REF]. Moreover, in the bacteria Pseudomonas fluorescens, [START_REF] Chrzanowski | Element content of Pseudomonas fluorescens varies with growth rate and temperature: A replicated chemostat study addressing ecological stoichiometry[END_REF] demonstrated that temperature can influence their elemental content regardless of their resource nutrient content, which appears consistent with the present results. This imbalance in carbon and nitrogen fluxes in P. miliaris between 17 and 20°C may have major implications for physiological processes, such as growth and reproduction [START_REF] Frost | The stoichiometry of dissolved organic carbon, nitrogen, and phosphorus release by a planktonic grazer, Daphnia[END_REF][START_REF] Heflin | Effect of diet quality on nutrient allocation to the test and Aristotle's lantern in the sea urchin Lytechinus variegatus (Lamarck, 1816)[END_REF].

Some studies have shown that temperature increase may alter macroalgal palatability to grazers (Poore et al., , ), through changes in nutritional qualities or levels of deterrent secondary metabolites [START_REF] Staehr | Physiological responses of Ecklonia radiata (Laminariales) to a latitudinal gradient in ocean temperature[END_REF][START_REF] Sudatti | Effects of abiotic factors on growth and chemical defenses in cultivated clones of Laurencia dendroidea J. Agardh (Ceramiales, Rhodophyta)[END_REF]. In this context, further labeling experiments should be conducted to examine the trophic transfer from macroalgae to grazers at different temperatures. In addition, these results highlight that quantitative estimation of trophic transfers between the two first trophic levels should not be based solely on carbon fluxes as is often the case.

In conclusion, the use of dual stable isotope labeling shows promise for testing the influence of various environmental factors on major macroalga-grazer trophic interactions within marine habitats, including their response to climate change. Further similar experimental studies should also couple stable isotope data related to assimilated nutrients with other physiological processes associated with feeding, such as ingestion, excretion or respiration.

  Individuals ofPsammechinus miliaris were collected from a maerl bed in the bay of Brest, France (48°17′18″N, 04°23′46″W) in April 2016 using a naturalist's dredge (width: 1 m, height: 0.2 m, net: 1.5 m long). Organisms were then transported in seawater tanks to the Roscoff Marine Station. To mitigate the stress experienced by the species during sampling and transport, organisms were maintained in open-flow aquaria pending experiments. Thalli of the seaweed Solieria chordalis were collected from maerl beds in the Bay of Brest (48°19′56″N, 04°19′56″W) in early June 2016 and transported to the Roscoff Marine Station.

  Team, 2014). δ 13 C and δ 15 N of labeled Solieria chordalis were compared between the beginning (0 day) and the end (8 days) using the non-parametric Wilcoxon-Mann-Whitney test, as assumptions of normality (Shapiro test) and homogeneity of variances (Bartlett test) were not met. The differences in δ 13 C and δ 15 N between labeled and non-labeled urchins were also examined using the Wilcoxon-Mann-Whitney test. The effects of temperature and time on δ 13 C and δ 15 N of urchins supplied with labeled S. chordalis were examined using the non-parametric Scheirer-Ray-Hare test, as assumptions of normality and homogeneity of variances (Bartlett test) were not met. Similarly, the Scheirer-Ray-Hare test was performed to examine the effect of increased temperature and time on urchin 13 C and 15 N uptake.

1

  Variations in δ 13 C (a) and δ 15 N (b; mean ± SE) in Psammechinus miliaris maintained for 6, 15 and 30 days at 17°C (black dots) and at 20°C (white dots) and supplied with labeled (full lines) and non-labeled (dotted lines) Solieria chordalis (n = 4) miliaris supplied with labeled S. chordalis exhibited 15 N specific uptake values ranging from 0.04 µg 15 N (day 30) to 0.11 µg 15 N (day 6)

  and respiration rates as the temperature increases (Cáceres-Puig, F I G U R E 2 Variations in 13 C (a) and 15 N (b) uptake (I 13 C and I 15 N, respectively; mean ± SE) in Psammechinus miliaris maintained for 6, 15 and 30 days at 17°C (black dots) and at 20°C (white dots) and supplied with labeled Solieria chordalis (n = 4)

  TA B L E 2 Results of Scheirer-Ray-Hare tests on the effect of temperature (17 and 20°C) and time (days 6, 15 and 30) and their combined effect on 13 C and 15 N uptake (I 13 C and I 15 N, respectively) in Psammechinus miliaris supplied with labeled Solieria chordalis (n = 4) Results of Scheirer-Ray-Hare tests to assess the effect of temperature (17 and 20°C) and time (days 6, 15 and 30) and their combined effect on the δ 13 C and δ 15 N in Psammechinus miliaris supplied with labeled Solieria chordalis(n = 4)

	TA B L E 1 Factor	df	δ 13 C (‰) H	p	δ 15 N (‰) H	p			I 13 Cµg 13 C g DW -1	I 15 Nµg 15 N g DW -1
	Temperature	1	7.3	.007**	1.2	.27	Factor	df	H	p	H	p
	Time	2	1.3	.53	4.6	.10
	Temperature × time	2	1.0	.60	2.2	.32	Temperature	1	5.8	.016*	0.74	.39
							Time	2	2.3	.32	11.9	.003**
	Significant p-values are in bold. *p < .05; **p < .01; ***p < .001.					Temperature × time	2	1.0	.62	1.5	.47
							Significant p-values are in bold.
							*p < .05; **p < .01; ***p < .001.
							DW, dry weight.

ACK N OWLED G EM ENTS

The authors thank the Center of Marine Biologic Resources at the Station Biologique de Roscoff for its kind permission to use their premises for the duration of the experiments. We are grateful to Olivier Bohner and Jérôme Coudret for their field and laboratory assistance and for the help in system maintenance. We acknowledge the crew of the research vessel Albert Lucas for their help with species collection. We also thank the SOMLIT (Institut National des Sciences de l'Univers -Centre National de la Recherche Scientifique) program for the temperature data sets provided. This work was supported by the Brittany Regional Council, the French National Research Agency via the 'Investment for the Future' program IDEALG (no. ANR-10-

Funding information Brittany Regional Council; French National Research Agency, Grant/Award Number: ANR-10-BTBR-04; Écosphère Continentale

S U PP O RTI N G I N FO R M ATI O N

Additional supporting information may be found online in the Supporting Information section at the end of the article.