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lambda语句摘要：mbda语句的定义和用途mbda语句的基本语法mbda语句在编程中的应用场景4.如何在不同编程语言中使用Lambda语句mbda语句的优缺点6.总结正文：Lambda语句是一种简洁的匿名函数，它允许你在不定义完整函数的情况下，仅实现特定功能。
Lambda语句在许多编程语言中都有应用，如Python、Java、C#等。
本文将详细介绍Lambda语句的定义、语法、应用场景以及在不同编程语言中的使用方法。
mbda语句的定义和用途Lambda语句，又称为匿名函数，是一种简洁的函数表示形式。
它通常由一个表达式和一对大括号组成。
Lambda语句的用途是在不定义完整函数的情况下，实现特定功能。
这种语句可以提高代码的可读性和简洁性，特别是在需要频繁使用相似功能的情况下。
mbda语句的基本语法Lambda语句的基本语法如下：```lambda arguments: expression```其中，`arguments`是传递给Lambda函数的参数，`expression`是Lambda函数的返回值。
需要注意的是，Lambda语句只能包含一个表达式，不能包含多条语句。
mbda语句在编程中的应用场景Lambda语句在编程中的应用场景主要包括：- 作为列表排序和过滤的函数，如Python中的`sorted()`和`filter()`函数的参数；- 作为函数参数，实现高阶函数（如Python的`map()`、`filter()`和`sorted()`函数）；- 实现特定算法，如快速排序算法中的分治策略。
4.如何在不同编程语言中使用Lambda语句在不同编程语言中，Lambda语句的使用方法略有不同。
以下是一些常见编程语言中的Lambda语句示例：- Python：```pythonlambda x, y: x + y```- Java：```javaList<Integer> numbers = Arrays.asList(1, 2, 3, 4, 5);List<Integer> squares = numbers.stream().map(n -> n * n).collect(Collectors.toList());```- C#：```csharpvar numbers = new List<int> { 1, 2, 3, 4, 5 };var squares = numbers.Select(n => n * n).ToList();```mbda语句的优缺点Lambda语句的优点：- 简洁明了，提高代码可读性；- 易于编写和调试；- 可以在不定义完整函数的情况下，实现特定功能。
lambda表达式的运算符-回复lambda表达式是一种简洁而强大的函数定义方式，它引入了一些特殊的运算符以帮助我们更灵活地定义函数。
本文将以中括号内的内容为主题，详细介绍lambda表达式中的运算符，并逐步回答相关问题。
首先，我们需要了解lambda表达式的基本结构。
一个lambda表达式由关键字lambda、参数列表、冒号和函数体组成，它的一般形式为：lambda 参数列表: 函数体。
lambda表达式可以定义匿名函数，它们不需要通过def关键字定义，也不需要给函数命名，因此更加简洁和直观。
[lambda表达式的运算符]:1. 算术运算符：- 加法：+- 减法：-- 乘法：*- 除法：/- 取模：- 幂运算：这些算术运算符在lambda表达式中的使用与普通的函数定义一致，可以进行加、减、乘、除、取模和幂运算等基本计算操作。
下面我们将通过示例来展示它们的用法。
示例1：lambda表达式中使用加法运算符pythonadd = lambda x, y: x + yprint(add(2, 3)) 输出5示例2：lambda表达式中使用除法运算符pythondivide = lambda x, y: x / yprint(divide(10, 2)) 输出5.0示例3：lambda表达式中使用取模运算符pythonmod = lambda x, y: x yprint(mod(10, 3)) 输出12. 比较运算符：- 相等：==- 不等：!=- 大于：>- 小于：<- 大于等于：>=- 小于等于：<=比较运算符常用于条件判断，用于比较两个值的大小或相等性，并返回相应的布尔值。
在lambda表达式中，我们可以使用这些比较运算符进行条件判断，根据比较结果返回不同的值。
下面是一些示例：示例4：lambda表达式中使用相等运算符pythonis_equal = lambda x, y: x == yprint(is_equal(2, 2)) 输出Trueprint(is_equal(2, 3)) 输出False示例5：lambda表达式中使用不等运算符pythonnot_equal = lambda x, y: x != yprint(not_equal(2, 2)) 输出Falseprint(not_equal(2, 3)) 输出True示例6：lambda表达式中使用大于等于运算符pythongreater_than_equal = lambda x, y: x >= yprint(greater_than_equal(2, 2)) 输出Trueprint(greater_than_equal(2, 3)) 输出False3. 逻辑运算符：- 与：and- 或：or- 非：not逻辑运算符用于进行逻辑运算，根据条件的真假返回相应的布尔值。
lambda表达式的exceptionLambda表达式是一种匿名函数，它可以用来创建简单的函数或者函数对象。
它的语法非常简洁，但在使用它时，我们需要特别注意异常处理。
在这篇文章中，我将为您详细介绍如何在Lambda表达式中处理异常，以及一些相关的最佳实践。
一、什么是Lambda表达式在介绍异常处理之前，让我们先回顾一下Lambda表达式的基本概念。
Lambda表达式是一种匿名函数，它可以在需要函数对象的地方使用，例如作为参数传递给其他函数或者存储在变量中。
它的语法由以下三个部分组成：1. 形参列表：包括函数的输入参数，多个参数之间用逗号隔开。
2. 箭头操作符：由一个短横线和一个大于号组成，表示函数的输入参数映射到函数体。
3. 函数体：包括一条或多条语句，表示函数的实现逻辑。
Lambda表达式的基本语法如下：[捕获列表] (参数列表) -> 返回类型{函数体}二、异常处理的重要性在编写程序时，异常处理是一项非常重要的任务。
当代码执行过程中发生异常时，如果没有适当的异常处理机制，程序可能会崩溃或者产生不可预期的结果。
因此，在使用Lambda表达式时，我们需要特别关注异常处理，以确保程序的稳定性和可靠性。
三、在Lambda表达式中处理异常的方法那么，在Lambda表达式中如何处理异常呢？以下是一些常用的方法：1. 使用try-catch语句：与传统的方法没有太大区别，可以在Lambda 表达式的函数体内部使用try-catch语句来捕获和处理异常。
例如：javaConsumer<String> consumer = (s) -> {try {...可能会抛出异常的代码...} catch (Exception e) {异常处理逻辑}};在上面的例子中，我们通过try-catch语句捕获Lambda表达式中可能抛出的异常，并在catch块中处理异常。
2. 使用声明异常的函数式接口：如果Lambda表达式抛出的是已经声明的异常，我们可以使用函数式接口，由调用方来处理异常。
lambda表达式格式以及应用场景
Lambda表达式的格式为：
```
lambda 参数列表: 表达式
```
其中，参数列表是一组逗号分隔的参数，冒号后面是一个表达式。
Lambda表达式可以有多个参数，也可以没有参数。
Lambda表达式的应用场景主要有以下几个：
1. 函数式编程：Lambda表达式可以用于函数式编程中的各种
函数，如map、filter、reduce等。
可以将Lambda表达式作为
参数传递给这些函数，并在函数内部进行调用，简化代码结构。
2. 排序：Lambda表达式可以用于排序算法中，作为比较函数
来指定排序规则。
3. 事件处理：在事件驱动的编程模型中，可以使用Lambda表
达式作为事件处理函数，简化事件处理的代码逻辑。
4. 线程与并发编程：在线程和并发编程中，可以使用Lambda
表达式来定义线程的执行逻辑，简化线程的创建和管理过程。
5. GUI编程：在图形用户界面编程中，可以使用Lambda表达
式来处理各种事件的响应函数，简化事件处理的代码。
Lambda表达式的应用场景不仅限于以上几个例子，根据实际
需求，还可以在很多其他场景中使用Lambda表达式来简化代
码的编写。
c++ 拉姆达表达式
C++中的Lambda表达式是一种匿名函数，它可以捕获外部变量并在其中执行代码。
使用Lambda表达式可以简化代码并提高代码的可读性和可维护性。
Lambda表达式的语法类似于函数，可以在函数内部定义Lambda 表达式。
Lambda表达式由方括号“[]”、参数列表、箭头符“->”和函数体组成。
方括号中可以包含Lambda表达式需要捕获的外部变量，可以使用“&”表示按引用捕获，也可以使用“= ”表示按值捕获。
Lambda表达式的参数列表是可选的，可以省略。
如果需要参数，可以在参数列表中指定参数的类型和名称。
箭头符“->”用于指定Lambda表达式的返回类型，如果Lambda 表达式没有返回值，则可以省略箭头符和返回类型。
函数体可以是任何合法的C++代码，可以包含任意数量的语句和变量定义。
Lambda表达式可以返回任何类型的值，包括void类型。
Lambda表达式的使用非常灵活，可以作为参数传递给函数、存储在变量中或作为函数的返回值。
在使用Lambda表达式时，需要注意Lambda表达式的生命周期，避免出现悬空指针等问题。
总之，Lambda表达式是C++中非常强大的特性，可以大大提高程序的灵活性和可读性。
掌握Lambda表达式的基本语法和用法对于C++程序员来说是非常重要的。
- 1 -。
lambda表达式的用法Lambda表达式的用法Lambda表达式是Java 8中引入的一个新特性，它是一种匿名函数，可以作为参数传递给方法或存储在变量中。
Lambda表达式的出现使得Java语言更加简洁、灵活，同时也提高了代码的可读性和可维护性。
本文将介绍Lambda表达式的用法。
1. Lambda表达式的语法Lambda表达式的语法非常简洁，它由三个部分组成：参数列表、箭头符号和函数体。
下面是一个Lambda表达式的示例：```(int x, int y) -> x + y```其中，参数列表为`(int x, int y)`，箭头符号为`->`，函数体为`x + y`。
Lambda表达式的参数列表可以为空，也可以有多个参数，多个参数之间用逗号分隔。
函数体可以是一个表达式，也可以是一个代码块。
如果函数体是一个表达式，则可以省略花括号和return关键字；如果函数体是一个代码块，则必须用花括号括起来，并且需要使用return关键字返回值。
2. Lambda表达式的类型推断Lambda表达式的类型推断是Java 8中的另一个新特性。
在Lambda表达式中，可以省略参数类型和返回值类型，编译器会根据上下文自动推断出类型。
例如，下面的代码中，编译器可以自动推断出Lambda表达式的参数类型为String，返回值类型为int：```Function<String, Integer> f = s -> s.length();```在Lambda表达式中，如果参数列表中只有一个参数，可以省略括号。
例如，下面的代码中，Lambda表达式的参数列表只有一个参数，可以省略括号：```Consumer<String> c = s -> System.out.println(s);```3. Lambda表达式的应用Lambda表达式可以应用于各种场景，例如集合操作、线程池、GUI编程等。
现代西班牙语第二册课后习题答案第1课II. 把动词变为陈述式过去未完成时1. cogía2.saludaba3.Asistiamos4.Madrugaba5. apeabas6.llegaba7.detenía8.entendías9.despedía 10.Jugábamosv.1. tenia cultivaba2.conversaban3.nadabamos4.podía5.asistian se divertian6.se alojaban7.solían era8.tenían era9.era queria 10.estabaVI.1. Mientras curazaba la plaza, me encontré con una amiga.2. El vistante pregunta si habia un momumento a Cervantes en el parque.3. Me dice que con estecarnet podia yo visitar gratis la exposicion.4. Nos pregunta dónde se conseguía planos de ciudad.5. Los turistas creen que las cosas les van a resultar muy complicadas.6. Nos señala aquel palacio que es inmenso.7. Como el museo de Prado es inmenso, no podemos recorrerlo en una mañana.8. El taxita nos lleva a un hostal que cerca de una estación de metro.9. Ella ve a alguien que camina hacia ella.10. No puedo asistir a la clase porque me siento indispuesto.VIII:. el ejercicio hecho el pescado cocinadola tema conversada la cosa perdidola nota copiado los libros repartidolos coches producidos el camino indicadola celebracion celebrado el monumento admiradoel huerto cultivado la entrada conseguidoel cuento contado la ropa metidoel invitado alojado en mi casaIX1. Quitan2. Me voy 3;ir 4.quitar 5.tocar 6.se fue 7.toca 8.toca 9.quito 10.Pon 11.Ponganse 12.sigue 13.Seguiste 14.pones 15.seguíaXI.1. que2.a3. de que a4.que a5.que6.A de7.de8.en con9.de a 10.a enXII.1. Haga el favor de pasarme las revistas que están en la mesa.2. El muchacho que nada en el rio es mi primo.3. Les voy a presentar a la señorita Gómez que estudió junto conmigo en una escuela hace varios años.4. Ve a ayudar a tu madre que barre el comedor.5. Puede usted pasarme el diccionario que tiene a su lado?6. Es inreresante el cuento que lee?7. Comemos en ese comedor que quede cerca de la bibioteca.8. Vamos a visitar a la profesora extranjera que trabaja en nuestra facultad.9. Levantó un viento que me secó el sudor.10. El doctor le curó la enfermedad el recetó una medicina tradicionalXV.Ana se lavantótarde ayer, porque se sentióindispuesto：le dolía la cabeza y la garganta, tenía la nariz tapada y perdióel apetito.Parecía que tenía fiebre. Se puso el termómentro：37.8grados. Decidióir al hospital. El médico le atenciósimpaticamente, le auscultó, le tomó el pulso, le tomóla temperatura y miró su garganta y lengua. Luego le dijo que era un resfriado, pero no muy grave y que no tenía por qué preocuparse.Ana fue a la farmaria con la receta que el médico le rectó. La enfermera todavía le puso una inyección. En casa ella desansaba unos días y se sentía mejor.第2课II1. Se celebraba una velada cada fin de semana el año pasado.2. El hombre cesaba el trabajo cada vez que oía el ruido.3. Los niños difrutaban mucho jugando todos las tardes durante las vacaciones.4. Todos se fijaban en mí siempre que etraba.5. Bailabamos cueca,marinera y tango en todas las fiestas en aquel entonces.6. Le aplaudíamos mucho todas las vecesque recitaba.7. Sudaba mucho jugando al boloncesto en quella época.8. Se despedían de nosotro a la diez todas las noches entonces.9. El médico me recetaba medicina tradicional china siempre que me atendia.10. La madre se preocupaba mucho por la hija porque regresaba tarde cadi todas las neches en aquella época.III.1. Dijo hacía2.preguntósentía3.Dijo dolía4.dije tenía5.entré acababa6.dijo tenia era7.Dijo podía8.Ví bailaba9.contestaba sabía 10.preguntaron gustabaIV.1. Me preguntó si me importaba esperar un rato.2. Dijiste que un asno pasataba tranquilamente en el prado.3. Contestaron que sentían mucho miedo.4. Le pregunté por qué se fingía enfermo.5. Repondió que el hombre que buscó cojeaba un poco.6. le dié que tenía razón.7. Le preguntamos que había un monumento a Cervantes en la plaza.8. Me preguntó dónde se podían conseguir entradas.9. Contestó que quería ver una exposicion de Goya.10. Dijieron que era muy complicado el problema.。
lambda 等价的方法引用lambda表达式和方法引用都是Java8中引入的新特性，可以用于简化代码。
在某些情况下，lambda表达式和方法引用可以互相替换使用，因为它们实际上是等价的。
Lambda表达式是一种匿名函数，可以将其视为一个代码块，可以作为参数传递给方法或存储在变量中。
方法引用是对现有方法的引用，可以像lambda表达式一样传递给方法或存储在变量中。
当lambda 表达式只是调用现有方法时，我们可以使用方法引用来代替它。
方法引用有四种类型：1. 静态方法引用：对于静态方法，我们可以使用类名和方法名来引用该方法。
例如，Math类的sqrt()方法可以使用以下静态方法引用：Math::sqrt2. 实例方法引用：对于实例方法，我们可以使用对象引用和方法名来引用该方法。
例如，字符串的length()方法可以使用以下实例方法引用：String::length3. 构造方法引用：对于构造方法，我们可以使用类名和new关键字来引用该方法。
例如，下面是一个构造方法引用，用于创建一个空的字符串： String::new4. 超类方法引用：对于超类方法，我们可以使用super关键字和方法名来引用该方法。
例如，Object类的toString()方法可以使用以下超类方法引用： super::toStringLambda表达式和方法引用的等价性可以通过以下代码证明：Lambda表达式：Function<String, Integer> lambdaFunc = s -> s.length();方法引用：Function<String, Integer> methodRef = String::length;这两个代码片段是等价的，都是将字符串的长度作为输入并返回它的长度(integer)。
因此，两种方法都可以使用，具有相同的功能效果。
总之，lambda表达式和方法引用是Java 8中非常强大的新特性，可以帮助我们在代码中减少冗余，提高代码的可读性和可维护性。
lambda 表达式8种原型-回复Lambda表达式是一种精简的函数写法，它可以在一行代码中定义和使用函数。
在Python中，Lambda表达式通常用于函数式编程，可以将其作为参数传递给其他函数，或者将其用于定义简单的匿名函数。
有8种不同的Lambda表达式原型，每种原型都有自己的特点和适用场景。
本文将一步一步回答关于这8种原型的问题，详细介绍它们的用法和示例。
1. 无参数原型：lambda: expression无参数原型的Lambda表达式不需要任何输入参数。
它只是简单地执行一个表达式，并返回结果。
这种原型的Lambda表达式通常用于执行简单的操作或返回固定的数值。
示例：x = lambda: 42print(x()) # 输出: 422. 单参数原型：lambda parameter: expression单参数原型的Lambda表达式接受一个输入参数，并通过表达式计算并返回结果。
这种原型的Lambda表达式通常用于需要对单个元素进行处理的函数。
示例：x = lambda a: a * 2print(x(5)) # 输出: 103. 多参数原型：lambda parameter1, parameter2, ...: expression多参数原型的Lambda表达式可以接受多个输入参数，并通过表达式计算并返回结果。
这种原型的Lambda表达式通常用于需要对多个元素进行处理的函数。
示例：x = lambda a, b: a + bprint(x(3, 4)) # 输出: 74. 默认参数原型：lambda parameter=default_value: expression默认参数原型的Lambda表达式在定义时可以给输入参数设置默认值。
如果调用Lambda表达式时未传入参数，则会使用默认值进行计算。
示例：x = lambda a=1, b=2: a + bprint(x()) # 输出: 3print(x(3)) # 输出: 55. 可变长度参数原型：lambda *args: expression可变长度参数原型的Lambda表达式可以接受任意数量的参数，并将它们作为一个元组传递给表达式。
万方数据万方数据万方数据爱因斯坦宇宙常数Λ的几何解释作者：曹盛林， Cao Shenglin作者单位：北京师范大学天文学天文系,北京,100875刊名：北京石油化工学院学报英文刊名：JOURNAL OF BEIJING INSTITUTE OF PETRO-CHEMICAL TECHNOLOGY年，卷(期)：2009，17(3)被引用次数：0次1.Goldhaber G.Perlmutter S A study of 42 type Ia supernovae and a resulting measurement of Omega(M) and Omega(Lambda) 1998(1-4)2.Chern S S Finsler GeometryIs Just Riemannian Geometry without the Quadratic Restriction 1996(09)3.Arnold V I Catastrophe Theory 19864.Asanov G S Finsler Geometry,Relativity and Gauge Theories 19855.Cao Shenglin The Theory of Rlativity and Super-Luminal Speeds Ⅰ 19886.Cao Shenglin The Theory of Rlativity and Super-Luminal Speeds Ⅱ 19907.Cao Shenglin The Theory of Rlativity and Super-Luminal Speeds Ⅲ 19928.Cao Shenglin The Theory of Rlativity and Super-Luminal Speeds Ⅵ 19929.Cao Shenglin The Theory of Rlativity and Super-Luminal Speeds V 199310.Balan V.Brinzei N Einstein equations for (h;v)-Berwald-Moor relativistic models 2006(01)1.学位论文吴剑锋静态de Sitter时空中洛伦兹破缺的电动力学2006最近，天文学家通过对极远处超新星的光度的观测[1，2，3，4]，推算出宇宙在各个不同时期的膨胀速率，发现宇宙在加速膨胀并且加速度基本保持不变。
a r X i v :a s t r o -p h /0201034v 1 3 J a n 2002Mon.Not.R.Astron.Soc.000,000–000(0000)Printed 1February 2008(MN L A T E X style file v1.4)Do Type Ia Supernovae prove Λ>0?Michael Rowan-Robinson1AstrophysicsGroup,Imperial College London,Blackett Laboratory,Prince Consort Road,London SW72BW1February 2008ABSTRACTThe evidence for positive cosmological constant Λfrom Type Ia super-novae is reexamined.Both high redshift supernova teams are found to underestimate the effects of host galaxy extinction.The evidence for an absolute magnitude-decay time relation is much weakened if supernovae not observed before maximum light are excluded.Inclusion of such objects artificially supresses the scatter about the mean relation.With a consistent treatment of host galaxy extinction and elimination of supernovae not observed before maximum,the evidence for a positive lambda is not very significant (3-4σ).A factor which may contribute to apparent faintness of high z supernovae is evolution of the host galaxy extinction with z.The Hubble diagram using all high z distance estimates,including SZ clusters and gravitational lens time-delay estimates,does not appear inconsistent with an Ωo =1model.Although a positive Λcan provide an,albeit physically unmotivated,resolution of the low curvature implied by CMB experiments and evidence that Ωo <1from large-scale structure,the direct evidence from Type Ia supernovae seems at present to be inconclusive.Key words:infrared:cosmology:observations1INTRODUCTIONThe claim that the measured brightnesses of Type Ia supernovae at redshifts 0.1-1.0imply Λ>0(Schmidt et al 1998,Garnavich et al 1998,Riess et al 1998,Perl-mutter et al 1999,Fillipenko and Riess 2000,Riess et al 2001,Turner and Riess 2001)has had a dramatic effect on cosmology.The model with λo =0.7,whereλo =Λ/3H 2o ,and Ω0=0.3has become a concensus model,believed to be consistent with most evidence from large-scale stucture and CMB fluctuations.In this paper I test the strength of the evidence that Λ>0and show that there are inconsistencies in the way the supernovae data have been analyzed.When these are removed,the strength of the evidence for Λ>0is much diminished.To set the scene,Fig 1shows B max versus log (cz)for 117Type Ia supernovae since 1956from the Barbon et al (1998)catalogue (excluding those la-belled ’*’which are discovery magnitudes only),to-gether with published supernovae from the high zprogrammes,corrected for Galactic and internal ex-tinction,but not for decay-time effects,together withpredicted curves from an Ωo =1model.At first sight there is not an enormous difference between the high z and low z supernovae,except that the latter seem to show a larger scatter.Fig 2shows the same exclud-ing less reliable data (flagged ’:’,in the Barbon et al catalogue,or objects with pg magnitudes only (Lei-bundgut et al (1991)),correcting for peculiar velocity effects (see section 3),using the Phillips et al (1999)internal extinction correction (see section 2)where available,and deleting two objects for which the dust correction is >1.4mag.The scatter for the low z su-pernovae appears to have been reduced.Finally Fig 3shows the supernovae actually used by Perlmutter et al (1999).Now the scatter in the low z supernovae is not much different from the high z supernovae and a difference in absolute magnitude between low z and high z supernovae,relative to an Ωo =1model,can be perceived.However comparison with Fig 2suggests that the low z supernovae used may be an abnormallyc0000RAS2Rowan-Robinson M.2345610152025Figure 1.B max versus log (cz)for all Type Ia super-novae post 1956.Magnitudes are corrected for Galactic extinction,and for internal extinction (using the de Vau-couleurs prescription -see section 2).A mean (B-V)colour of 0.0has been assumed for the supernovae for which only V at maximum light is known.Solid curves are loci for Ωo =1model,with M B =−19.64,−17.44(H o =100),corresponding to <M B >±2σfor solution (1)of Table 2.luminous subset of all supernovae.We will return to this point in section 4.Excellent recent reviews of Type Ia supernovae,which fully discuss whether they can be thought of as a homogenous population,have been given by Branch (1998),Hildebrand and Niemeyer (2000)and Lei-bundgut (2000,2001).In this paper I shall assume that they form a single population and that their ab-solute magnitude at maximum light depends,at most,on a small number of parameters.I do not,for exam-ple,consider the possibility of evolution,discussed by Drell et al (2000).Absolute magnitudes are quoted for H o =100throughout.2INTERNAL EXTINCTIONOne of the most surprising claims of the high z super-novae teams is that internal extinction in the high z supernovae is small or even negligible (Perlmutter et al 1999,Riess et al 2000).While some nearby Type Ia supernovae take place in elliptical galaxies,where internal extinction in the host galaxy may indeed be negligible,the majority take place in spiral galaxies,where internal extinction can not be neglected.More-over as we look back towards z =1,we know from the CFRS survey that there is a marked increase in the fraction of star-forming systems (Lilly et al 1996),and2345610152025Figure 2.B max versus log (cz)for Type Ia supernovae post 1956,excluding less reliable data,correcting for effect of peculiar velocity,and using the Phillips et al (1999)internal extinction correction where available.Solid curves as for Fig 1.2345610152025Figure 3.B max versus log (cz)for Type Ia supernovae used by Perlmutter et al (1999).No correction for internal extinction is applied.Solid curves are as for Fig 1,but shifted by +0.33mag.to account for non-correction for extinction.c0000RAS,MNRAS 000,000–000Do Type Ia Supernovae proveΛ>0?3 we would expect the average host galaxy extinctionto be,if anything,higher than in low z supernovae.On average,the extinction along the line of sightto a(non edge-on)spiral galaxy can be representedby de Vaucouleurs’s prescription(de Vaucouleurs etal1976):A int=0.17+α(T)lg10(a/b),for T>-4,=0forT=-4,-5,where T is the de Vaucouleurs galaxy type,a,bare the major and minor diameters of the galaxy,andα(T)=0.2,0.4,0.6,0.7,0.8for T=-3,-2,-1,0,1-8,respectively.I assume lg10(a/b)=0.2where this notknown.The extinction to a particularly supernova can beexpected to show marked deviations from this aver-age value,since the dust distributions in galactic discsare very patchy.Because of the cirrus-like distributionof interstellar dust clouds,lines-of-sight to some starswill have much lower extinctions than this average value.The presence of dense molecular clouds in the galaxy means that other lines-of-sight can have very much higher extinctions.Phillips et al(1999)have an-alyzed the extinction to62Type Ia supernovae,using both the colours at maximum light and the colours at very late time,30-90days after maximum.Their extinction corrections do resolve a number of cases of anomalously faint Type Ia supernovae(eg1995E, 1996Z,1996ai).The agreement of their internal ex-tinction values with those given by the de Vaucouleurs prescription is not brilliant in detail(see Table1), but as expected give broadly the same median values (median A int for Phillips et al sample:0.29,median de Vaucouleurs correction for same sample:0.33). These median values are broadly consistent with the Monte Carlo simulations of Hatano et al(1998).Fig-ure4shows the correlation between A int and the(B-V)colour at maximum light,corrected for Galactic extinction,with different symbols for Phillips et al (1999)estimates and those derived from the de Vau-couleurs prescription.The distribution is consistent with an intrinsic(unreddened)colour range of(B-V) =-0.1to0.1,combined with the usual A int=4.14 (B-V)relation.Riess et al(1998)have given estimates of the to-tal extinction(A int+4.14E(B−V)Gal for their sample of low supernova derived both via the MLCS method and via a set of templates(their Table10).We can compare the template estimates directly with those of Phillips et al(1999)for the same galaxies(Fig5). The Riess et al values are lower on average by0.22 magnitudes,which implies that the set of templates (and the training set for the MLCS method)have not been completely dereddened.The large scatter in this diagram perhaps indicates the difficulty of estimating the host galaxy extinction for supernovae.If this aver-age underestimate of0.22mag.is added to the Riess et al estimates of A int for high-z supernovae,the aver-age value of A int for these is almost identical to thatB-V00.20.40.51Figure4.A int versus(B−V)(corrected for Galactic extinction)using Phillips et al(1999)extinction correction filled circles)or de Vaucouleurs prescription(open circles). The solid lines show the dust reddening loci for intrinsic (unreddenened)colours of E(B-V)=-0.1and0.1.for low z supernovae.Thus the claim that high z su-pernovae have lower extinction than low z supernovae seems to be based on a systematic underestimate of extinction in the high z galaxies.Perlmutter et al(1999)estimate that host galaxy extinction is on average small both in local and high z supernovae,and neglect it in most of their solutions. Parodi et al(2000)neglect internal extinction com-pletely,preferring to cut out the redder objects(B-V>0.1)from their samples.This will not change the relative absolute magnitudes between low and high z (or between supernovae with Cepheid calibration and the others)provided the two samples end up with the same mean extinction.This,however,might be diffi-cult to guarantee.I have preferred to correct the low z supernovae as described above,and then correct the Perlmutter et al data by an average host galaxy ex-tinction of0.33mag.Ellis and Sullivan(2001)have carried out HST imaging and Keck spectroscopy on host galaxies for supernova used by Perlmutter et al(1999)andfind that the Hubble diagram for supernovae in later type galaxies shows more scatter than those hosted by E/S0galaxies,presumably due to the effects of host galaxy extinction.3CORRECTION FOR PECULIARVELOCITY EFFECTSFor nearby supernovae it is important to correct for the peculiar velocity of the host galaxy.Hamuy et alc 0000RAS,MNRAS000,000–0004Rowan-RobinsonM.Figure 5.Difference between the total extinction es-timated by Riess et al (1998)and the sum of the host galaxy extinction estimated by Phillips et al (1999)and the Galactic extinction,plotted against ∆m 15.(1995)tackle this be setting a minimum recession ve-locity of 2500km/s for their local sample.However since peculiar velocities can easily be in excess of 500km/s this is probably not sufficient for an accurate result.Here I have used the model for the local (z <0.1)velocity field developed to analyze the CMB dipole,using data from the IRAS PSCz redshift sur-vey (Rowan-Robinson et al 2000).I estimate the ve-locity error in estimates from this model to be 100+0.2x V km/s for V <15000km/s ,400km/s for V >15000km/s .These errors are incorporated into the subsequent analysis (points are weighted by error −2).The code for this peculiar velocity model is available at /∼mrr.Tables 1and 2gives data for the Type Ia super-novae used in the present study.The columns are as follows:(1)supernova name,(2)host galaxy reces-sion velocity,(3)same,corrected for peculiar veloc-ity,(4)blue magnitude at maximum light B max ,(5)E (B −V )Gal ,from Schlegel et al (1998),(6)A int from de Vaucouleurs et al (1976)presciption,(7)A int from Phillips et al (1999),(8)absolute magnitude M B (Ωo =1universe),(9)distance modulus,assuming M B =-19.47(Gibson et al 2000),(10)∆m 15,(11)(B −V )o .Sources of B max ,in order of preference:Hamuy et al (1996)/Perlmutter et al (1999)/Riess et al (1998,Barbon et al (1999).Sources of ∆m 15,in order of pref-erence:Phillips et al (1999),Saha et al (1999),Riess et al (1998),Parodi et al (2001).Sources of (B −V )o :Hamuy et al (1996),Saha et al (1999),Parodi et al (2001),Leibundgut et al (1991).4ABSOLUTE MAGNITUDE-DECAY TIME RELATIONOnce we have corrected for the effects of internal ex-tinction,we can test whether the absolute magnitude of supernovae at maximum light depends on decay time,or some other parameter like the colour at maxi-mum light (B −V )o ,as proposed by Tripp and Branch (1999),and Parodi et al (2000).Phillips (1993)proposed that there is a strong dependence of absolute magnitude at maximum light and the decay-time,characterized by the blue mag-nitude change during the 15days after maximum,∆m 15.Specifically he found dM B /d ∆m 15= 2.70.Riess et al (1995)showed that the dispersion in the sn Ia Hubble diagram was significantly reduced by ap-plying this correction.Tammann and Sandage (1995)used supernovae in galaxies for which distance es-timates were available to set a strong limit on the slope of the M B −∆m 15relation (<0.88).Hamuy et al (1996)used a new sample of well-studied super-novae to derive a B-band slope of 0.784±0.18for the M B −∆m 15relation,consistent with the Tammann and Sandage limit,and much lower than the original claim of Phillips (1993).Riess et al (1996)have dis-cussed a related method of analyzing this correlation,the MLCS method.However there is a further consideration here.It is really only valid to carry out this analysis on su-pernovae which have been detected prior to maximum light.The process used hitherto by all workers in this field of extrapolating to maximum light assuming an M B −∆m 15,or equivalently MLCS,relation is assum-ing that all extrapolated objects adhere to the mean line of the relation,thus underestimating the scatter in the relation.This is a process which artificially im-proves the apparent signal-to-noise of the final Hub-ble relation or Λ>0signal.Hamuy et al (1995)do make some allowance for this in assigning a larger un-certainty (and hence lower weight)to supernova first observed after maximum.Fig 6shows a plot of the absolute magnitude at maximum light,M B ,corrected by 0.784∆m 15,versus (B −V )o ,the colour at maximum light corrected for the effects of extinction,using the Phillips et al (1999)estimates of extinction.No clear correlation remains between these corrected quantities.Thus correction for the M B −∆m 15relation removes most of the cor-relation between M B and (B −V )o .Fig 7shows the relations between M B ,corrected for extinction,and ∆M 15,for objects detected at least 1day prior to maximum light.The best fit linear relation is shown,which has slope 0.99±0.38,con-sistent with values reported by Hamuy et al (1995),0.85±0.13,and by Hamuy et al (1996),0.78±0.18.However the significance of the relation is reduced,because of the smaller number of data points,and is now only 2.6σ.The rms deviation from this mean relation is 0.44mag,much larger than is generallyc0000RAS,MNRAS 000,000–000Do Type Ia Supernovae prove Λ>0?5Figure 6.M B at maximum light,corrected for extinction (using Phillips et al prescription)and by 0.784∆m 15,versus (B −V )o .claimed for this relation.For example,Riess et al (1996)claim that the residual sigma after correction for the M B −∆m 15relation is 0.12mag.The spurious reduction in scatter generated by extrapolating supernovae first observed after maxi-mum can be seen in Fig 8,which shows the same re-lation for supernovae first observed after maximum.The same effect can in fact be seen in the top panel of Fig 4in Parodi et al (2000).The marked difference between Fig 7and 8suggests that supernovae first observed after maximum should be excluded from the analysis.Perlmutter et al (1999)have applied a different version of the M B −∆m 15method,which they call the ’stretch’method,to high z supernovae.This method appears to give a significantly lower correction to M B as a function of ∆m 15(Leibundgut 2000).Figure 9shows the Perlmutter et al ’stretch’correction to the absolute magnitude,as a function of ∆m 15,for 18low z supernovae (note that they omit the two supernovae with ∆m 15=1.69from their solution).The slope is 0.275±0.04,only one third of the Hamuy et al (1996)value.The method has been further discussed by Efstathiou et al (1999)and Goldhaber et al (2001),but no explanation for the lower slope is given.5TESTING FOR POSITIVE ΛWith the criteria established above (i)that a full and consistent correction must be made for extinction,(ii)that if the the M B −∆m 15relation is used it should only be applied to supernovae detected beforemaxi-Figure 7.M B at maximum light,corrected for extinction,versus ∆m 15,for supernovae first observed before maxi-mum.The best-fitting line has slope -0.99,but the data are also consistent with nocorrelation.Figure 8.M B at maximum light,corrected for extinc-tion,versus ∆m 15,for supernovae not observed until after maximum.Note the (spurious)reduction in scatter.mum light,we can now reexamine the Hubble diagramfor supernovae.I consider several different samples:(1)all well-observed supernovae,with no correc-tion for the M B −∆m 15relation.Supernovae with only pg magnitudes are excluded,as also are super-novae first observed after maximum.The de Vau-couleurs extinction correction is used for supernovaec0000RAS,MNRAS 000,000–0006Rowan-RobinsonM.Figure 9.Perlmutter et al ’stretch’correction to M B ,∆P erl ,versus ∆m 15,for low z supernovae.The best fit slope is 0.275,well below the value of 0.784quoted by Hamuy et al (1996).not studied by Phillips et al (1999).Supernovae with A int >1.4are excluded.(2)all well-observed supernovae for which in addition ∆m 15is known,with correction for the M B −∆m 15relation,using the 0.784slope of Hamuy et al (1966).(3)as (2),but with internal extinction set to zero,as advocated by Perlmutter et al (1999).(4)as (2),but using the Perlmutter ’stretch’cor-rection (or 0.275∆m 15where stretch correction not available).(5)as (2)but using the quadratic ∆m 15correc-tion of Phillips et al (1999).It was not possible to independently check the effect of applying the MLCS method for correcting for decay-time correlations because the set of training vectors published by Riess et al (1996)is not the one actually being used in the high z supernova analysis (Riess et al 1998).The mean absolute magnitudes for low z (z <0.1)and high z supernovae,in an Einstein de Sitter model are tabulated for the different samples in Table 3.Without the correction for ∆m 15,the significance of the difference in absolute magnitude for 53low and 52high redshift supernovae is only 2.8σ,hardly suffi-cient to justify the current wide acceptance of positive Λ.The significance increases to 3.5σif only the 26supernovae observed since 1990are used.Including the ∆m 15correction,the significance increases to 4.0σ(3.9σif we use the Perlmutter cor-rection,4.6σif we use the quadratic ∆m 15correction of Phillips et al 1999,which is the form used by Riesset al 1998).Could this increase in significance be due to some bias in the local supernova sample for which ∆m 15has been measured ?The mean absolute mag-nitude at maximum light,corrected for extinction but not for ∆m 15,for all 53’good’low z supernovae is -18.54.The mean for those for which ∆m 15has been measured is -18.64,0.1magnitudes brighter.In fact the difference in the mean between those for which ∆m 15has been measured and those for which it has not been measured is 0.23mag.,almost the size of the whole signal on which the claim for positive Λis based.However there is no difference in mean abso-lute magnitude between the mean absolute magnitude for the 10local Calan Tololo supernovae used by by the high-z supernova teams and the 16other local su-pernovae studied since 1990,so there is no evidence that the Calan Tololo sample is biased.An alternative explanation of the fainter mean absolute magnitudes seen in the pre-1990data is a systematic error in the photographic photometry used.There is another factor which may contribute to the apparent faintness of high z supernovae.Current models of the star formation history of the universe,which show a strong peak in the star formation rate at z =1-2,imply that the mean dust optical depth in galaxies would be expected to increase with red-shift out to z =1.5-2(Calzetti and Heckman 1999,Pei et al 1999).Line 6of Table 3shows the effect of adopting Calzetti and Heckman’s model 3for the evolution of A int (z ),a model which agrees well with models for infrared and submillimetre sourcounts and backgrounds (eg Rowan-Robinson 2001a)and direct evidence from oberved estimates of the star forma-tion history (Rowan-Robinson 2001b).This correc-tion,which increases the average A int for the high z supernovae by 0.15mag.,is sufficient to reduce the significance of the magnitude difference between high and low z supernovae,in an Einstein de Sitter model,to 3.0and 1.5σfor cases with and without correction for ∆m 15.6HUBBLE DIAGRAM AT HIGH REDSHIFTRowan-Robinson (2001c)has reviewed distance esti-mates and the Hubble constant.In addition to Type Ia supernovae,S-Z clusters and gravitational lens time-delay methods also give distance estimates at z >0.1.Figure 10shows a compilation of all these high z estimates.I have also included the HDF super-nova (1997ff)at the redshift 1.7,proposed by Riess et al (2001)although Rowan-Robinson (2001b)finds a photometric redshift z =1.4for the parent galaxy.The assumed B band magnitude at maximum light is 25.5(Fig 7of Riess et al)and the mean absolute magnitude for Type Ia supernovae is assumed to be -19.47(Gibson et al 2000).The model with Ωo =1is a good overall fit to these data and the HDF supernovac0000RAS,MNRAS 000,000–000Do Type Ia Supernovae proveΛ>0?7lies right on theΩo=1mean line.A least squaresfit to these data,with an assumed H o of63km/s/Mpc (Rowan-Robinson2001c)and the spatial curvature parameter k=0,yieldsΩo=0.81±0.12,where the error in the distance modulus has been taken to be 0.35mag.for all points.7DISCUSSION AND CONCLUSIONS (1)I have reanalyzed the evidence that high-z super-novae support a universe with positiveΛ.(2)Both high-z supernova teams appear to have underestimated host galaxy extinction.(3)The evidence for an M B−∆m15relation is weaker than previously stated(only2.6σ)if analysis is restricted to supernovae observed before maximum. The rms deviation about the mean relation is signifi-cantly larger than previously claimed.(4)After consistent corrections for extinction are applied the significance of the difference in absolute magnitude between high and low z supernovae,in an Einstein de Sitter(Ωo=1)universe,is2.8-4.6σ,de-pending whether(and how)the M B−∆M15correc-tion is applied,so such a model can not really be rejected conclusively by the present data.(5)The Hubble diagram based on all high red-shift estimates supports an Einstein de Sitter uni-verse.The HDF-N supernova favours such a universe also,contrary to the published claims of Riess et al (2001).(6)The community may have been too hasty in its acceptance of a positiveΛuniverse,for which no physical motivation exists,and needs to reconsider the astrophysical implications of the more natural Einstein de Sitter,Ωo=1,model.For the supernova method,the need is to continue study of low z su-pernovae to improve understanding of extinction and of the absolute magnitude decay-time relation,and to consider shifting towards infrared wavelengths,as advocated by Meikle(2000),in order to reduce the effects of extinction.Of course the arguments presented here do not prove thatΛ=0.The combination of the evidence from CMBfluctuations for a spatiallyflat universe with a variety of large-scale structure arguments for Ωo=0.3-0.5may still make positiveΛmodels worth pursuing.However it would seem to be premature to abandon consideration of other alternatives.8ACKNOWLEDGEMENTSI thank Peter Meikle,Adam Riess,Steve Warren, Martin Haehnelt,Bruno Leibundgut,David Branch, Eddie Baron,Richard Ellis and an anonymous ref-eree for helpful comments or discussions,and Mark Phillips and Bernhard Parodi for supplying machine-readable versions of their data.REFERENCESBarbon R.,Buondi V.,Cappellaro E.,Turatto M.,1999, AAS139,531Branch D.,1998,ARAA36,17Branch D.,Romanishin W.,and Baron E.,1996,ApJ465, 73Calzetti D.and Heckman T.M.,1999,ApJ519,27Drell P.S.,Loredo 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1.110.211974J7468739015.600.06920.23--19.2734.55--0.281975G1900224614.440.00920.29--17.6533.58--1975N1867162014.000.03020.28--17.4633.06-0.171976J4558416414.280.02630.56--19.4933.080.900.021978E4856442315.400.16120.33--18.8333.87--0.191979B954169912.700.00990.19--18.6831.94-0.271980N1806144212.500.02070.210.20-18.5831.67 1.280.051981B1804126611.740.02070.430.45-19.3130.69 1.10-0.021981D1806144212.590.02070.21--18.5031.76 1.280.191983G1172151112.970.01980.36--18.3732.00-0.011983U1156148313.400.02160.29--17.8432.49--1983W1937217113.300.01040.52--18.9532.21--1986A3039338514.400.04090.20--18.6233.50--1987D2227255113.700.02170.33--18.7632.75--1988F5274509014.800.02230.25--19.0833.93--1989A2514291414.100.02210.17--18.4933.31--1989B72661712.340.03040.41 1.36-18.1031.27 1.31-1989M1518126612.560.03760.24--18.3531.63--1990N998126612.760.02430.280.37-18.2331.85 1.070.041990Y108001080417.700.00950.000.94-18.4737.13 1.130.331990af151801480717.870.03360.250.16-18.3136.95 1.560.051991M2169256214.550.04610.33--17.8933.63--1992A1845148412.560.01380.250.00-18.3631.72 1.470.021992G1580219313.630.01210.38--18.4132.77--1992P7616815916.080.02100.330.29-18.8735.130.87-0.031992ag7544772416.410.08560.330.41-18.8135.20 1.190.081992al4377471114.600.03150.330.04-18.9433.61 1.11-0.051992bc5700560215.160.01960.330.00-18.6734.220.87-0.081992bh135001338417.700.02170.330.49-18.5436.75 1.050.081992bo5575524515.860.02700.250.00-17.8634.97 1.690.011992bp237902379118.410.07170.210.00-18.8137.37 1.32-0.051993L1925178413.200.01790.66--18.6232.11 1.18-1993O153001475017.670.06100.210.00-18.4536.68 1.22-0.091993ag147001541717.720.10310.210.29-18.9636.55 1.320.031994D450126611.840.02170.370.00-18.7630.85 1.32-0.041994S4550452314.800.01700.330.00-18.5633.87 1.100.011994ae1282148313.200.02840.340.49-18.2732.210.860.101995D1967244213.400.05330.250.16-18.9232.400.99-0.021995al1541201113.360.02070.340.61-18.8632.400.83-1996X2032181113.220.06850.000.04-18.4032.41 1.25-0.031996bl108001006317.050.11730.330.33-18.8035.70 1.170.041996bo5241461716.170.07240.25 1.15-18.6135.09 1.25-1997ej6686677415.850.01700.25--18.6435.00--1998bu943148312.220.03030.28 1.35-20.1131.28 1.01-University of Texas PressTammann G.A.and Sandage A.,1995,ApJ452,16Tripp R.and Branch D.,1999,ApJ525,209Turner M.S.and Riess A.G.,2001,ApJL(in press),astro-ph/0106051de Vaucouleurs G.,de Vaucouleurs A.,Corwin H.G.Jr,1976,Second Reference Catalogue of Bright Galaxies,c 0000RAS,MNRAS000,000–000。
