## What are the sets on which norm-closedness implies weakly closedness?

Let $$X$$ be a Banach space. Let $$C$$ be a convex, and normed-closed subset of $$X$$. It is well-known that $$C$$ becomes weakly closed subset of $$X$$. I want to know is there any well-know class of non convex sets which has this property?

i.e., a class of sets in $$X$$, not necessary convex on which, norm-closed implies weakly-closed.

## What’s the exact consistency strength of this very elegant axiom system for classes and sets?

Notation: Let $$\phi$$ be any formula in $$\small \sf FOL \small(=,\in, W)$$; let $$\varphi$$ be any formula in $$\small\sf FOL \small (=,\in)$$ having $$x$$ free, and whose parameters are among $$x_1,..,x_n$$.

Define: $$elm(y)\iff \exists z (y \in z)$$

Where $$“elm”$$ is short for “..is an element”

Axioms:

1.Extensionality: $$\forall z (z \in x \leftrightarrow z \in y) \to x=y$$

2.Class Comprehension: $$\exists x \forall y (y \in x \leftrightarrow elm(y) \land \phi)$$

3.Set Comprehension: $$x_1,..,x_n \in W \to [ \forall x (\varphi \to x \subseteq W) \leftrightarrow \forall x (\varphi \to x \in W)]$$

4.Foundation: $$x \neq \emptyset \to \exists y \in x (y \cap x = \emptyset)$$

5.Choice over all classes.

The basic two axioms of this theory are the two schemes, all the rest of axioms are indeed interpretable from them. $$W$$ stands for the class of all sets, that’s why the predicate $$elm$$ is used here instead of the usual denotation of it as $$set$$ in Morse-Kelley class theory.

In my own personal opinion, this kind of axiomatization is ideal, I think it’s of the most elegant ones. There is no dispute over axioms 1,4,and 5. Schema 2 is the most natural principle about classes. Schema 3 technically sums up all of what’s in standard set theory in a neat manner. All axioms are very clean and fairly natural.

Question: What’s the exact consistency strength of this theory?

## Sufficient sets of colimits in small categories

Let $$C$$ be a small category, and consider the class of diagrams $$G:D\to C$$, with $$D$$ a small category, that have colimits in $$C$$. This is a proper class even when $$C$$ is very small, e.g. whenever $$D$$ has a terminal object $$t$$, any functor $$G:D\to C$$ has a colimit $$G(t)$$, and there is a proper class of small categories with a terminal object.

However, those colimits feel kind of “trivial”; in some cases at least we can find a small set of diagrams that “carry all the nontrivial information” about colimit diagrams in $$C$$. For instance, if $$C$$ is a poset, then it suffices to consider injective functors $$G$$ (and we may as well take $$D$$ to be discrete as well), and these form an essentially small set. For a non-posetal $$C$$ we can’t restrict to injective functors, since coproducts are not idempotent, but maybe there is some other restriction that works. Note that by Freyd’s theorem, a non-posetal small category does have a bound on the cardinality of coproducts that it can admit; but this doesn’t quite answer the question itself, since a particular colimit can exist even if the coproducts that would be necessary to construct it from coproducts and coequalizers do not.

Here are two ways to make the question precise:

1. Given a small category $$C$$, is there a small set $$L$$ of diagrams $$G:D\to C$$ with colimits such that for any diagram $$G’:D’\to C$$ with a colimit, there is a $$(D,G)\in L$$ and a final functor $$F:D\to D’$$ such that $$G = G’ \circ F$$?

2. Given a small category $$C$$, is there a small set $$L$$ of diagrams $$G:D\to C$$ with colimits such that if a functor $$H:C\to E$$ preserves colimits of all diagrams in $$L$$, then it preserves all colimits that exist in $$C$$?

Any solution to question 1 is also a solution to question 2, but I’m not sure whether the converse holds. The mention of Freyd’s theorem above suggests that a solution might require classical logic — I would find it more surprising if such a set existed for a non-posetal small complete category, although I don’t immediately see an argument that it cannot.

One can of course also ask similar questions for enriched categories, internal categories, $$\infty$$-categories, and so on. Bonus points go to an answer that applies more generally in such contexts.

## A conjecture concerning symmetric convex sets

Let’s suppose that $$S \subset \mathbb{R}^n$$ is convex and symmetric so:

$$$$x \in S \iff -x \in S \tag{1}$$$$

Now, if we define the radius of $$S$$ as $$R$$ such that:

$$$$R = \sup_{x \in S} \lVert x \rVert \tag{2}$$$$

and use (2) to define:

$$$$V = \{x \in S: \lVert x \rVert = R\} \tag{3}$$$$

then I conjecture that:

$$$$S = \text{conv}(V) \tag{*}$$$$

I have worked out special cases of this problem within the context of high-dimensional probability but I suspect that it’s generally true.

Might there be a theorem which guarantees this result?

## Merging data sets

In a google sheet I have multiple tables containing a list of dates. I want to create a single (sorted) list containing all the dates and the table they came from.

Table foo: 1.1.19 3.1.19  Table bar: 2.1.19 4.1.19  Result: 1.1.19 foo 2.1.19 bar 3.1.19 foo 4.1.19 bar 

Is this somehow possible with regular formulas or do I need to switch to custom functions here?

## “Please wait while Bamboo sets up your instance” taking for ever on Mac

I am trying to install and configure Bamboo on mac. I have installed it and now at the stage of running the setup wizard. After giving the license and selecting “Express installation”, now am stuck at “Please wait while Bamboo sets up your instance” page. It’s taking forever. What could be causing this?

Updating the errors I found in the logs:

2019-06-14 11:32:06,547 WARN [setupDefaultGeneralBackgroundThread]         [GenericTypeAwarePropertyDescriptor] Invalid JavaBean property          'mappingResources' being accessed! Ambiguous write methods found next to actually used [public void   bucket.core.persistence.hibernate.ConfigurableLocalSessionFactoryBean.setMappingResources(bucket.core.persistence.hibernate.MappingResources)]: [public void org.springframework.orm.hibernate5.LocalSessionFactoryBean.setMappingResources(java.lang.String[])] 2019-06-14 11:32:06,902 INFO [http-nio-8085-exec-13] [AccessLogFilter] 0:0:0:0:0:0:0:1 POST http://localhost:8085/setup/setupDefaultGeneral.action 422910kb 2019-06-14 11:32:06,977 WARN [setupDefaultGeneralBackgroundThread] [XmlWebApplicationContext] Exception encountered during context initialization - cancelling refresh attempt: org.springframework.beans.factory.BeanCreationException: Error creating bean with name 'reloadableBeanFactories' defined in class path resource [applicationContextPlugin.xml]: BeanPostProcessor before instantiation of bean failed; nested exception is org.springframework.beans.factory.BeanCreationException: Error creating bean with name 'org.springframework.transaction.config.internalTransactionAdvisor': Cannot resolve reference to bean 'org.springframework.transaction.annotation.AnnotationTransactionAttributeSource#0' while setting bean property 'transactionAttributeSource'; nested exception is org.springframework.beans.factory.BeanCreationException: Error creating bean with name 'org.springframework.transaction.annotation.AnnotationTransactionAttributeSource#0': BeanPostProcessor before instantiation of bean failed; nested exception is org.springframework.beans.factory.BeanCreationException: Error creating bean with name 'customBuildCompleteActionAdvisor' defined in class path resource [applicationContextManagers.xml]: Cannot resolve reference to bean 'hibernateInterceptor' while setting bean property 'advice'; nested exception is org.springframework.beans.factory.BeanCreationException: Error creating bean with name 'hibernateInterceptor' defined in class path resource [applicationContextPersistence.xml]: Cannot resolve reference to bean 'sessionFactory' while setting bean property 'sessionFactory'; nested exception is org.springframework.beans.factory.BeanCreationException: Error creating bean with name 'sessionFactory' defined in class path resource [applicationContextPersistence.xml]: Invocation of init method failed; nested exception is java.lang.NoClassDefFoundError: javax/xml/bind/JAXBException 2019-06-14 11:32:06,979 ERROR [setupDefaultGeneralBackgroundThread] [ContextLoader] Context initialization failed org.springframework.beans.factory.BeanCreationException: Error creating bean with name 'reloadableBeanFactories' defined in class path resource [applicationContextPlugin.xml]: BeanPostProcessor before instantiation of bean failed; nested exception is org.springframework.beans.factory.BeanCreationException: Error creating bean with name 'org.springframework.transaction.config.internalTransactionAdvisor': Cannot resolve reference to bean 'org.springframework.transaction.annotation.AnnotationTransactionAttributeSource#0' while setting bean property 'transactionAttributeSource'; nested exception is org.springframework.beans.factory.BeanCreationException: Error creating bean with name 'org.springframework.transaction.annotation.AnnotationTransactionAttributeSource#0': BeanPostProcessor before instantiation of bean failed; nested exception is org.springframework.beans.factory.BeanCreationException: Error creating bean with name 'customBuildCompleteActionAdvisor' defined in class path resource [applicationContextManagers.xml]: Cannot resolve reference to bean 'hibernateInterceptor' while setting bean property 'advice'; nested exception is org.springframework.beans.factory.BeanCreationException: Error creating bean with name 'hibernateInterceptor' defined in class path resource [applicationContextPersistence.xml]: Cannot resolve reference to bean 'sessionFactory' while setting bean property 'sessionFactory'; nested exception is org.springframework.beans.factory.BeanCreationException: Error creating bean with name 'sessionFactory' defined in class path resource [applicationContextPersistence.xml]: Invocation of init method failed; nested exception is java.lang.NoClassDefFoundError: javax/xml/bind/JAXBException     at org.springframework.beans.factory.support.AbstractAutowireCapableBeanFactory.createBean(AbstractAutowireCapableBeanFactory.java:490)     at org.springframework.beans.factory.support.AbstractBeanFactory.lambda$doGetBean$  0(AbstractBeanFactory.java:317)     at org.springframework.beans.factory.support.DefaultSingletonBeanRegistry.getSingleton(DefaultSingletonBeanRegistry.java:222)     at org.springframework.beans.factory.support.AbstractBeanFactory.doGetBean(AbstractBeanFactory.java:315)     at org.springframework.beans.factory.support.AbstractBeanFactory.getBean(AbstractBeanFactory.java:204)     at org.springframework.context.support.PostProcessorRegistrationDelegate.registerBeanPostProcessors(PostProcessorRegistrationDelegate.java:236)     at org.springframework.context.support.AbstractApplicationContext.registerBeanPostProcessors(AbstractApplicationContext.java:708)     at org.springframework.context.support.AbstractApplicationContext.refresh(AbstractApplicationContext.java:533)     at org.springframework.web.context.ContextLoader.configureAndRefreshWebApplicationContext(ContextLoader.java:409)     at org.springframework.web.context.ContextLoader.initWebApplicationContext(ContextLoader.java:291)     at com.atlassian.config.spring.BootstrappedContainerContext.refresh(BootstrappedContainerContext.java:22)     at com.atlassian.bamboo.setup.DefaultHibernateConfigurator.refreshSpringContextAndCreateDatabase(DefaultHibernateConfigurator.java:120)     at com.atlassian.bamboo.setup.DefaultHibernateConfigurator.configureDatabase(DefaultHibernateConfigurator.java:63)     at com.atlassian.config.bootstrap.DefaultAtlassianBootstrapManager.bootstrapDatabase(DefaultAtlassianBootstrapManager.java:258)     at com.atlassian.bamboo.ww2.actions.setup.SetupUtilityBean.setupStandardDatabase(SetupUtilityBean.java:187)     at com.atlassian.bamboo.ww2.actions.setup.SetupUtilityBean.setupEmbeddedDatabase(SetupUtilityBean.java:178)     at com.atlassian.bamboo.ww2.actions.setup.SetupDefaultsAction$SetupDefaultDatabase.call(SetupDefaultsAction.java:86) at com.atlassian.bamboo.ww2.actions.setup.SetupDefaultsAction$  SetupDefaultGeneral.call(SetupDefaultsAction.java:69)     at com.atlassian.bamboo.ww2.actions.setup.SetupDefaultsAction$SetupDefaultGeneral.call(SetupDefaultsAction.java:43) at com.atlassian.bamboo.ww2.actions.setup.SetupInvoker.execute(SetupInvoker.java:15) at com.atlassian.bamboo.ww2.actions.setup.SetupDefaultsAction.general(SetupDefaultsAction.java:40) at java.base/jdk.internal.reflect.NativeMethodAccessorImpl.invoke0(Native Method) at java.base/jdk.internal.reflect.NativeMethodAccessorImpl.invoke(NativeMethodAccessorImpl.java:62) at java.base/jdk.internal.reflect.DelegatingMethodAccessorImpl.invoke(DelegatingMethodAccessorImpl.java:43) at java.base/java.lang.reflect.Method.invoke(Method.java:566) at ognl.OgnlRuntime.invokeMethod(OgnlRuntime.java:899) at ognl.OgnlRuntime.callAppropriateMethod(OgnlRuntime.java:1544) at ognl.ObjectMethodAccessor.callMethod(ObjectMethodAccessor.java:68) at com.opensymphony.xwork2.ognl.accessor.XWorkMethodAccessor.callMethodWithDebugInfo(XWorkMethodAccessor.java:98) at com.opensymphony.xwork2.ognl.accessor.XWorkMethodAccessor.callMethod(XWorkMethodAccessor.java:90) at ognl.OgnlRuntime.callMethod(OgnlRuntime.java:1620) at ognl.ASTMethod.getValueBody(ASTMethod.java:91) at ognl.SimpleNode.evaluateGetValueBody(SimpleNode.java:212) at ognl.SimpleNode.getValue(SimpleNode.java:258) at ognl.Ognl.getValue(Ognl.java:470) at ognl.Ognl.getValue(Ognl.java:434) at com.opensymphony.xwork2.ognl.OgnlUtil$  3.execute(OgnlUtil.java:401)     at com.opensymphony.xwork2.ognl.OgnlUtil.compileAndExecuteMethod(OgnlUtil.java:453)     at com.opensymphony.xwork2.ognl.OgnlUtil.callMethod(OgnlUtil.java:399)     at com.opensymphony.xwork2.DefaultActionInvocation.invokeAction(DefaultActionInvocation.java:438)     at com.opensymphony.xwork2.DefaultActionInvocation.invokeActionOnly(DefaultActionInvocation.java:293)     at org.apache.struts2.interceptor.BackgroundProcess$1.run(BackgroundProcess.java:55) at java.base/java.lang.Thread.run(Thread.java:834) Caused by: org.springframework.beans.factory.BeanCreationException: Error creating bean with name 'org.springframework.transaction.config.internalTransactionAdvisor': Cannot resolve reference to bean 'org.springframework.transaction.annotation.AnnotationTransactionAttributeSource#0' while setting bean property 'transactionAttributeSource'; nested exception is org.springframework.beans.factory.BeanCreationException: Error creating bean with name 'org.springframework.transaction.annotation.AnnotationTransactionAttributeSource#0': BeanPostProcessor before instantiation of bean failed; nested exception is org.springframework.beans.factory.BeanCreationException: Error creating bean with name 'customBuildCompleteActionAdvisor' defined in class path resource [applicationContextManagers.xml]: Cannot resolve reference to bean 'hibernateInterceptor' while setting bean property 'advice'; nested exception is org.springframework.beans.factory.BeanCreationException: Error creating bean with name 'hibernateInterceptor' defined in class path resource [applicationContextPersistence.xml]: Cannot resolve reference to bean 'sessionFactory' while setting bean property 'sessionFactory'; nested exception is org.springframework.beans.factory.BeanCreationException: Error creating bean with name 'sessionFactory' defined in class path resource [applicationContextPersistence.xml]: Invocation of init method failed; nested exception is java.lang.NoClassDefFoundError: javax/xml/bind/JAXBException at org.springframework.beans.factory.support.BeanDefinitionValueResolver.resolveReference(BeanDefinitionValueResolver.java:378) at org.springframework.beans.factory.support.BeanDefinitionValueResolver.resolveValueIfNecessary(BeanDefinitionValueResolver.java:110) at org.springframework.beans.factory.support.AbstractAutowireCapableBeanFactory.applyPropertyValues(AbstractAutowireCapableBeanFactory.java:1597) at org.springframework.beans.factory.support.AbstractAutowireCapableBeanFactory.populateBean(AbstractAutowireCapableBeanFactory.java:1349) at org.springframework.beans.factory.support.AbstractAutowireCapableBeanFactory.doCreateBean(AbstractAutowireCapableBeanFactory.java:572) at org.springframework.beans.factory.support.AbstractAutowireCapableBeanFactory.createBean(AbstractAutowireCapableBeanFactory.java:495) at org.springframework.beans.factory.support.AbstractBeanFactory.lambda$  doGetBean$0(AbstractBeanFactory.java:317) at org.springframework.beans.factory.support.DefaultSingletonBeanRegistry.getSingleton(DefaultSingletonBeanRegistry.java:222) at org.springframework.beans.factory.support.AbstractBeanFactory.doGetBean(AbstractBeanFactory.java:315) at org.springframework.beans.factory.support.AbstractBeanFactory.getBean(AbstractBeanFactory.java:204) at org.springframework.aop.framework.autoproxy.BeanFactoryAdvisorRetrievalHelper.findAdvisorBeans(BeanFactoryAdvisorRetrievalHelper.java:91) at org.springframework.aop.framework.autoproxy.AbstractAdvisorAutoProxyCreator.findCandidateAdvisors(AbstractAdvisorAutoProxyCreator.java:109) at org.springframework.aop.aspectj.annotation.AnnotationAwareAspectJAutoProxyCreator.findCandidateAdvisors(AnnotationAwareAspectJAutoProxyCreator.java:92) at org.springframework.aop.aspectj.autoproxy.AspectJAwareAdvisorAutoProxyCreator.shouldSkip(AspectJAwareAdvisorAutoProxyCreator.java:101) at org.springframework.aop.framework.autoproxy.AbstractAutoProxyCreator.postProcessBeforeInstantiation(AbstractAutoProxyCreator.java:254) at org.springframework.beans.factory.support.AbstractAutowireCapableBeanFactory.applyBeanPostProcessorsBeforeInstantiation(AbstractAutowireCapableBeanFactory.java:1060) at org.springframework.beans.factory.support.AbstractAutowireCapableBeanFactory.resolveBeforeInstantiation(AbstractAutowireCapableBeanFactory.java:1033) at org.springframework.beans.factory.support.AbstractAutowireCapableBeanFactory.createBean(AbstractAutowireCapableBeanFactory.java:484) ... 42 more Caused by: org.springframework.beans.factory.BeanCreationException: Error creating bean with name 'org.springframework.transaction.annotation.AnnotationTransactionAttributeSource#0': BeanPostProcessor before instantiation of bean failed; nested exception is org.springframework.beans.factory.BeanCreationException: Error creating bean with name 'customBuildCompleteActionAdvisor' defined in class path resource [applicationContextManagers.xml]: Cannot resolve reference to bean 'hibernateInterceptor' while setting bean property 'advice'; nested exception is org.springframework.beans.factory.BeanCreationException: Error creating bean with name 'hibernateInterceptor' defined in class path resource [applicationContextPersistence.xml]: Cannot resolve reference to bean 'sessionFactory' while setting bean property 'sessionFactory'; nested exception is org.springframework.beans.factory.BeanCreationException: Error creating bean with name 'sessionFactory' defined in class path resource [applicationContextPersistence.xml]: Invocation of init method failed; nested exception is java.lang.NoClassDefFoundError: javax/xml/bind/JAXBException at org.springframework.beans.factory.support.AbstractAutowireCapableBeanFactory.createBean(AbstractAutowireCapableBeanFactory.java:490) at org.springframework.beans.factory.support.AbstractBeanFactory.lambda$  doGetBean$0(AbstractBeanFactory.java:317) at org.springframework.beans.factory.support.DefaultSingletonBeanRegistry.getSingleton(DefaultSingletonBeanRegistry.java:222) at org.springframework.beans.factory.support.AbstractBeanFactory.doGetBean(AbstractBeanFactory.java:315) at org.springframework.beans.factory.support.AbstractBeanFactory.getBean(AbstractBeanFactory.java:199) at org.springframework.beans.factory.support.BeanDefinitionValueResolver.resolveReference(BeanDefinitionValueResolver.java:367) ... 59 more Caused by: org.springframework.beans.factory.BeanCreationException: Error creating bean with name 'customBuildCompleteActionAdvisor' defined in class path resource [applicationContextManagers.xml]: Cannot resolve reference to bean 'hibernateInterceptor' while setting bean property 'advice'; nested exception is org.springframework.beans.factory.BeanCreationException: Error creating bean with name 'hibernateInterceptor' defined in class path resource [applicationContextPersistence.xml]: Cannot resolve reference to bean 'sessionFactory' while setting bean property 'sessionFactory'; nested exception is org.springframework.beans.factory.BeanCreationException: Error creating bean with name 'sessionFactory' defined in class path resource [applicationContextPersistence.xml]: Invocation of init method failed; nested exception is java.lang.NoClassDefFoundError: javax/xml/bind/JAXBException at org.springframework.beans.factory.support.BeanDefinitionValueResolver.resolveReference(BeanDefinitionValueResolver.java:378) at org.springframework.beans.factory.support.BeanDefinitionValueResolver.resolveValueIfNecessary(BeanDefinitionValueResolver.java:110) at org.springframework.beans.factory.support.AbstractAutowireCapableBeanFactory.applyPropertyValues(AbstractAutowireCapableBeanFactory.java:1597) at org.springframework.beans.factory.support.AbstractAutowireCapableBeanFactory.populateBean(AbstractAutowireCapableBeanFactory.java:1349) at org.springframework.beans.factory.support.AbstractAutowireCapableBeanFactory.doCreateBean(AbstractAutowireCapableBeanFactory.java:572) at org.springframework.beans.factory.support.AbstractAutowireCapableBeanFactory.createBean(AbstractAutowireCapableBeanFactory.java:495) at org.springframework.beans.factory.support.AbstractBeanFactory.lambda$  doGetBean\$  0(AbstractBeanFactory.java:317)     at org.springframework.beans.factory.support.DefaultSingletonBeanRegistry.getSingleton(DefaultSingletonBeanRegistry.java:222)     at org.springframework.beans.factory.support.AbstractBeanFactory.doGetBean(AbstractBeanFactory.java:315)     at org.springframework.beans.factory.support.AbstractBeanFactory.getBean(AbstractBeanFactory.java:204)     at org.springframework.aop.framework.autoproxy.BeanFactoryAdvisorRetrievalHelper.findAdvisorBeans(BeanFactoryAdvisorRetrievalHelper.java:91)     at org.springframework.aop.framework.autoproxy.AbstractAdvisorAutoProxyCreator.findCandidateAdvisors(AbstractAdvisorAutoProxyCreator.java:109)     at org.springframework.aop.aspectj.annotation.AnnotationAwareAspectJAutoProxyCreator.findCandidateAdvisors(AnnotationAwareAspectJAutoProxyCreator.java:92)     at org.springframework.aop.aspectj.autoproxy.AspectJAwareAdvisorAutoProxyCreator.shouldSkip(AspectJAwareAdvisorAutoProxyCreator.java:101)     at org.springframework.aop.framework.autoproxy.AbstractAutoProxyCreator.postProcessBeforeInstantiation(AbstractAutoProxyCreator.java:254)     at org.springframework.beans.factory.support.AbstractAutowireCapableBeanFactory.applyBeanPostProcessorsBeforeInstantiation(AbstractAutowireCapableBeanFactory.java:1060)     at org.springframework.beans.factory.support.AbstractAutowireCapableBeanFactory.resolveBeforeInstantiation(AbstractAutowireCapableBeanFactory.java:1033)     at org.springframework.beans.factory.support.AbstractAutowireCapableBeanFactory.createBean(AbstractAutowireCapableBeanFactory.java:484)     ... 64 more Caused by: org.springframework.beans.factory.BeanCreationException: Error creating bean with name 'hibernateInterceptor' defined in class path resource [applicationContextPersistence.xml]: Cannot resolve reference to bean 'sessionFactory' while setting bean property 'sessionFactory'; nested exception is org.springframework.beans.factory.BeanCreationException: Error creating bean with name 'sessionFactory' defined in class path resource [applicationContextPersistence.xml]: Invocation of init method failed; nested exception is java.lang.NoClassDefFoundError: javax/xml/bind/JAXBException     at org.springframework.beans.factory.support.BeanDefinitionValueResolver.resolveReference(BeanDefinitionValueResolver.java:378)     at org.springframework.beans.factory.support.BeanDefinitionValueResolver.resolveValueIfNecessary(BeanDefinitionValueResolver.java:110)     at org.springframework.beans.factory.support.AbstractAutowireCapableBeanFactory.applyPropertyValues(AbstractAutowireCapableBeanFactory.java:1597)     at org.springframework.beans.factory.support.AbstractAutowireCapableBeanFactory.populateBean(AbstractAutowireCapableBeanFactory.java:1349) mac bamboo 

## Iterating over a union of sets denoted by bitmasks

Consider the set $$\mathbb{B}^n$$ of all $$n$$-digit binary numbers. Let us define a bitmask as a tuple $$M=(m_0,\ldots,m_{n-1})$$, where $$m_i\in \{0,1,*\}$$. Such bitmask denotes a set $$S \subset \mathbb{B}^n$$ containing all numbers with digits $$b_0…b_{n-1}$$ such that $$\forall i \in [0, n-1]: m_i \neq * \Rightarrow b_i=m_i$$.

For example the bitmask 1*0* would denote the set $$\{1000, 1001, 1100, 1101\}$$.

Given a set of $$k$$ such bitmasks $$\{M_i: i \in [0, k-1]\}$$, how to efficiently iterate over the union of the corresponding sets of binary numbers in lexicographical order? By efficiently I mean doing some pre-processing in polynomial time and then computing each next number in $$O(n)$$ amortized time and also using a polynomial amount of memory.

## Inner fibrations are Kan fibrations on Map sets

Firstly, a bit of notation. Let $$C$$ be a simplicial set. We define, for $$x,y \in C$$ vertices in $$C$$

$$Map(x,y) = \{x\}\times_{\Delta^{\{0\}}}Map_{sSet}(\Delta^1,C) \times_{\Delta^{\{1\}}} \{y\}$$

the sSet of arrows from x to y. You can imagine a $$n$$ simplex here as a cylinder with basis $$\Delta^n$$, with all $$1_x$$ on the left base and all $$1_y$$ on the right base. From this description it is evident that $$Map(x,y) \simeq (C_{/y})_x$$, the cones over y with all $$1_x$$ at the basis. Symmetrically, it is equivalent to $$(C_{x/})_y$$.

As the two sSet are fibers of a right and a left fibration, they are respectively left inner fibrant and right inner fibrant. Thus, the map sets are inner fibrant.

I am searching to generalize this to an arbitrary inner fibration $$p:C \to D$$, and show that $$Map(x,y) \to Map(px,py)$$ is a Kan fibration.

Thanks!

## Defining computable functions on arbitrary sets

Turing machines take inputs that are strings of symbols from some alphabet, and they give outputs that are strings of symbols from the same alphabet. To show that a function is computable, we have to exhibit a Turing machine that computes it. In the case where the function is not from strings to strings, something about this bothers me.

I will introduce the problem with a simple example. Suppose that I define a function from binary trees to binary trees, and I want to show, formally, that it’s computable. To do that I would have to exhibit a Turing machine that computes it. Since binary trees are not strings, I first have to design a string representation of binary trees. For example, I could represent the tree

as ((*,*),(*)). Having done this, I can then pass binary trees to Turing machines in string form, and get trees back in string form, no problem.

But here’s my problem: the representation of binary trees as strings is itself a function (in fact a bijection) from binary trees to strings. How can I know formally that this function is a computable one?

On an intuitive level I have no problem seeing that it is, but on a formal level I can’t exhibit a Turing machine to verify it, because then I would need a string representation of binary trees, and we quickly get into an infinite regress. For simple things like binary trees the computability of the representation is obvious, but for more complicated sets it might not be.

Can this circle be squared — is there a formal definition of computability for functions of arbitrary countable sets that might not admit an obvious mapping to strings of symbols?

Here is another way of explaining my question: we might choose to define a model of computation that operates inherently on trees or some other set rather than strings. Indeed, many such models of computation have been proposed, including the lambda calculus and combinator calculi (which are arguably more naturally seen as operating on trees rather than strings), the recursive functions (which operate on integers), and various cellular automata (which operate on infinite strings or two dimensional lattice configurations, rather than finite strings). It’s only really historical accident that led the string-based Turing machines to become the gold standard against which other models of computation are judged.

The problem is that without being able to talk about computable mappings between these sets, it’s not strictly formally possible to say whether these other models of computation are equivalent to Turing machines or not. While it usually doesn’t raise any practical problems to just think of everything in terms of string representations, it seems dissatisfying that our theory of computation doesn’t treat all models of computation equally, but instead privileges those whose input and output are from a particular kind of set.

So really my question is whether a theory of computation exists has been formulated that is model-agnostic, in precisely this sense of not privileging one particular kind of set for input and output. It’s quite an abstract question, motivated more by a desire for formal niceness than by any practical concerns.

One suspects, for example, that the category theorists might have done some work in this direction. This is because category theory tends to talk about the properties of functions without talking much about the sets they map, so a “category of computable functions” would probably not know or care whether its underling objects were sets of strings or something else. This is an approach I would be particularly keen to read about if it exists. (I’ve asked another question, specifically about the category theory aspect, over at MathOverflow.)

## The maximum difference between the number of elements in the two sets of equal length of consecutive numbers that divisible by some prime numbers

Suppose that $$p_1,…,p_k$$ are distinct prime numbers.

Let $$f(n,l)$$ be equal to the number of elements from set $$\{n+1,n+2,…,n+l\}$$ that are divisible by some $$p_1,…,p_k$$. Is it true that $$\sup_{n,m,l \in \mathbb{N}} |f(n,l)-f(m,l)| = O(k).$$