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curious little things - aggregated feedsenMath Overflow Recent Questions: Erdos multiplication table problem in $\mathbb{Z} / p\mathbb{Z}$
https://mathoverflow.net/questions/301080/erdos-multiplication-table-problem-in-mathbbz-p-mathbbz
<p>Let $p$ be a prime, let $k\geq2$ and let $ (2p)^{\frac{1}{k}} \leq N << p$.
Let $A_N = \{\bar{a} : 1 \leq a \leq N\}$, where $\bar{a} := a + p\mathbb{Z} \in \mathbb{Z} / p\mathbb{Z}$. Consider $A_N^k = \{ \bar{a}_1\cdot...\cdot\bar{a}_k : \bar{a}_i \in A_N\ $for all $ i\}$.<br>
What is known about $|A_N^k| $ ?<br>
I am especially interested in asymptotic (in $p$) lower bounds for when $N=C \cdot p^{\frac{1}{k}+\epsilon}$ or $N=C \cdot p^{\frac{1}{k}}\cdot \log ^c p$ (other than the lower bounds inherited from the original multiplication table problem for $N^\prime = p^{\frac{1}{k}} < N$).</p>Thu, 24 May 2018 15:58:35 -0600Math Overflow Recent Questions: Mathematical modeling of cross advertising between inet blogs (or sites or TamTam channels )
https://mathoverflow.net/questions/301079/mathematical-modeling-of-cross-advertising-between-inet-blogs-or-sites-or-tamta
<p>Let me try to describe what nowdays happens in some part of inet blogging, it seems to me that there might be intersting math behind that. Might be some game theory approach should be used since on the one hand blogs are competing with each other for subscribers (=money), on the other hand they should act cooperatively in order to increase number of subscribers (making cross advertising).
Hope some equilibrium between competition-cooperation might exists providing optimal solution. The question would be what is that optimal "strategy".</p>
<hr>
<p><strong>Setup:</strong></p>
<p>There are inet blogs, each blog has $K_i$ subscribers (=readers).</p>
<p>Blog earns $K_i/( \sum K_i )$ money.</p>
<p>So: more subsribers blog has (relatively to other blogs) - more it earns. So the goal is it increase number of subscribers. </p>
<p>Assume the following cooperation behaviour: 10 blogs make agreement and each of them recommend to its subscribers to subscribe to that 10 blogs. Assume some small percent e.g. 1% of subscribers follow that recommendation, so $.01*(K_2+K_3+...K_{10}) $ subscribers comes to blog $1$ from blogs 2,3,...10, and
$.01*(K_1+K_3+...K_{10})$ comes to blog 2 from blogs 1,3,4...10 and so on. </p>
<p><strong>Question:</strong> What should be optimal choice of 9 partners for your blog, if you want to maximize your profit ? Assume you can offer cooperation to any 9 blogs
and they have right to accept / not accept your offer.</p>
<p>What is the point here:
if you are the largest subscriber holder - probably you should NOT cooperate with anybody since it will decrease your relative advantage,
on the other hand smaller bloggers will cooperate between themselves and if you do nothing sooner or later they will overcome you.
That is why you probably should also act. </p>
<p>We might assume that cooperation proccess happens at the end of each day.
And money are paid at the end of each day.
So there is time evolution. Probably at the end we will come to uniform distribution of subscribers between blogs.</p>
<hr>
<p>It might be model should include more assumptions to be more interesting/realistic. We should take into account that if two blogs have the same audience, then self-advertising will not help.</p>
<p><strong>Question:</strong> Any improvements to the setup ?</p>
<hr>
<p>What blogging type I am talking about.
That happened recent 1-2 years in messenger Telegram, where blogs called "channels". Top channels now up to 500K subcribers and like a small mass-media.
Bloggers often use the trick described above to increase there audience.
Now another messenger TamTam is doing the same and you are welcome to math blog: <a href="https://tamtam.chat/math20" rel="nofollow noreferrer">https://tamtam.chat/math20</a></p>Thu, 24 May 2018 15:31:34 -0600Math Overflow Recent Questions: Universality of Singular Binary Quadratic Forms
https://mathoverflow.net/questions/301077/universality-of-singular-binary-quadratic-forms
<p>I have been working with singular forms modulo $p$, and it is easy to show that a singular binary quadratic form $Q$ cannot be universal mod $p$ (for p an even or odd prime) via Legendre symbol arguments or finite calculation. I have also found countless references showing conditions for universality of non-singular forms.</p>
<p>However, I want to know: Can a singular form $Q$ be universal over the $p$-adics, $\mathbb{Q}_p$?</p>Thu, 24 May 2018 14:58:11 -0600Math Overflow Recent Questions: Are there any statistical metrics that satisfy this kind of condition?
https://mathoverflow.net/questions/301076/are-there-any-statistical-metrics-that-satisfy-this-kind-of-condition
<p>Let $f=N(\mu,\sigma^2)$ be a univariate normal distribution with mean $\mu$ and variance $\sigma^2$ and let $f_1 = N(\mu+\epsilon,\sigma^2)$ and $f_2=N(\mu,(\sigma+\epsilon)^2)$ be some small perturbations to $f$. Are there any statistical metrics $D(\cdot,\cdot)$ (e.g. Kolmogorov-Smirnov, Wasserstein, Prokhorov, etc.) for which we can say something about the derivative $$\frac{d}{d\epsilon} D(f,f_1)$$ or $$\frac{d}{d\epsilon} D(f,f_2)$$ evaluated at $\epsilon=0$? How about if $f$ were, say, an exponential distribution instead?</p>Thu, 24 May 2018 14:57:24 -0600Math Overflow Recent Questions: Is there a name for a multidirectional graph that has functions associated with its edges instead of weights?
https://mathoverflow.net/questions/301075/is-there-a-name-for-a-multidirectional-graph-that-has-functions-associated-with
<p>Imagine a problem involving the fuel cost of driving a vehicle between multiple locations. Using a multidirectional graph with the nodes as locations, weights on edges could represent fuel costs. Now, imagine that there is a fuel loss associated with starting the engine of the vehicle, a fuel loss that does not depend on the location to which the vehicle is travelling. How might that information be encoded in the graph? Imagine there are other characteristics that might be encoded. How might these be encoded in an edge function as opposed to an edge weight?</p>
<p>What kind of graphs am I looking for here?</p>Thu, 24 May 2018 14:46:25 -0600Math Overflow Recent Questions: Asymptotic formula for absolute difference of number of prime factors between consecutive integers
https://mathoverflow.net/questions/301074/asymptotic-formula-for-absolute-difference-of-number-of-prime-factors-between-co
<p>For $ n=p_1^{\alpha_1}\cdots p_k^{\alpha_k}$ define $\Omega(n)= \alpha_1+\cdots+\alpha_k$.</p>
<p>What is known about the asymptotic behavior as $N\rightarrow\infty$ for sums of the form</p>
<p>$$\sum_{n=1}^N |\Omega(n+1)-\Omega(n)| \quad ?$$</p>Thu, 24 May 2018 14:41:46 -0600Math Overflow Recent Questions: dimension vector of indecomposable module over preprojective algebra
https://mathoverflow.net/questions/301072/dimension-vector-of-indecomposable-module-over-preprojective-algebra
<p>It is well-known that there are finitely many indecomposable module over the preprojective algebra associated to a quiver $Q$ if and only if $Q=A_2,A_3,A_4$ and tame type for $A_5$ and wild for others.
Now let say $Q$ is a ADE Dynkin diagram, and $V$ be an indecomposable preprojective algebra. What can we say about the dimension vector $|V|$ about $V$, does $|V| \in \mathbb{N}_0^I$ have an explicit bound?</p>
<p>I think for $Q=A_5$, for any indecomposable module the dimension vector of which is bounded by $(1,2,2,2,1)$.
What about other types?</p>Thu, 24 May 2018 14:34:04 -0600Math Overflow Recent Questions: What is the intuitive account about entities present in non-extensional models of set theory?
https://mathoverflow.net/questions/301069/what-is-the-intuitive-account-about-entities-present-in-non-extensional-models-o
<p>It is known that the standard foundational theory of mathematics $\text{ZFC}$ is reducible [interpreted in] to a proper fragment of it axiomatized by:</p>
<ol>
<li>Set Union</li>
<li>Power </li>
<li>Separation</li>
<li>Collection</li>
<li>Infinity</li>
</ol>
<p>Also Separation and Collection axiom schemas can be replaced by pairing and the following modified replacement schema, that I've worked on lately.</p>
<p>If $\phi(x,z)$ is a formula in which only symbols $``x,z"$ occur free, and those only occur free, then: $$\forall A \exists B \forall y [y \in B \leftrightarrow \exists x \in A \forall z (z \in y \leftrightarrow \phi(x,z))]$$</p>
<p>Another modified form of Replacement that even prove pairing and union is:</p>
<p>If $\phi(x,y)$ is a formula in which symbols $``x,y"$ occur free, and those only occur free, then all closures of the following are axioms.</p>
<p>$$[\forall x \exists z \forall y (\phi(x,y) \to y \in z)] \to \forall A \exists B \forall y (y \in B \leftrightarrow \exists x \in A (\phi(x,y)))$$</p>
<p>So this schema with Power and Infinity is equi-interpretable to ZFC.</p>
<p>Now my question is a little bit conceptual: Since this is enough to formalize most of mathematics, and since it would be consistent [should ZF be consistent], then what would be the entities that theory is formalizing? the absence of Extensionality in some sense permits non-extensional objects, but those cannot be sets, simply the term "non-extensional sets" is paradoxical, though used, sets are extensional, even some go so far to say that they must be well founded and hold that this is a part of the conception of what a set is, similarly some say choice is also a part of what constitute sets. Now this theory permit [in principle] violation of all of these. So what are those entities. During proofs concerned with equi-interpreting this with full ZFC, one do imaginatively conceive those entities as a kind of collections that has copies, so in some sense collectivity is part of the intuitive picture of those entities. Still those are not sets. </p>
<p><strong>Question:</strong> is there any known intuitive line of investigation of those entities? </p>
<p>Non-extensional models of ZFC are well known and they've been researched by Dana Scott and many others, so what was the intuitive picture about those entities.</p>
<p>One thing to be mentioned is that since the axiomatization is a fragment of ZFC, a theory about sets, then whatever that intuitive account about those entities is, it must be part of the intuitive account about sets.</p>
<p>I personally think that those entities might be "collectivity states of affairs", those need not be extensional, nor well founded, nor choice respective. The same set of objects can be collected by distinct collectivity states of affairs, the latter refers to a process of collectivity of the collection, for example two distinct persons might collect the same collection, but the states of affairs of collectivity of that collection are distinct, having different details, etc..
Can we replace the "is a member of" intuition with "is collected by" , collectivity need not respect foundation, since self collectivity is a possibility, it can be cyclical, it can go to infinity. Choice is the main axiom that gives the impression of being about collectivity rather than collection, since it carries the impression of an act, still imagining non-choice respective collectivity is justified. </p>
<p>Collectivity seems to be a weaker concept than a set, since the later is a collectivity state of affairs, but fairly specific and rigid one, one that is completely determined by what is collected. </p>
<p>Even the cumulative hierarchy of sets, seems to be about collectivity rather than the passive set concept. </p>
<p>So is it fair to say that most of mathematics is really grounded in collectivity state of affairs. Has such line of investigation been perused heavily before, especially in connection to non-extensional, non-well founded, non-choice related models of set theory?</p>Thu, 24 May 2018 14:11:50 -0600Math Overflow Recent Questions: What is spectral multiplicity for multiplication operators in general von Neumann algebra set up?
https://mathoverflow.net/questions/301068/what-is-spectral-multiplicity-for-multiplication-operators-in-general-von-neuman
<p>When two multiplication operators $M_{f}$ and $M_{g}$ acting on $L^2(X,\mu) $and $L^2(Y,\nu)$ are unitary equivalent? How multiplicity function look like here? What is the spectral multiplicity in this case and moreover is algebraic multiplicity replaced by cyclic vectors in the Hilbert space?. I know for bounded self-adjoint operators acting on $\mathcal{H}$ unitary equivalent when these have same spectrum and same multiplicity function. For an unbounded positive self-adjoint operator, can be possible to say it is unitary equivalent to unbounded function corresponding multiplication operator?</p>Thu, 24 May 2018 13:59:23 -0600Math Overflow Recent Questions: Is this expression always irrational?
https://mathoverflow.net/questions/301066/is-this-expression-always-irrational
<p>Is it right that</p>
<p><strong>$$\sqrt[a]{2^{2^n}+1}$$</strong></p>
<p>for every $$a>1,n \in \mathbb N $$ </p>
<p>is always irrational?</p>Thu, 24 May 2018 13:15:29 -0600