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### Can an upper bound for $r_{0}(n)$ be reached from a duality principle about the distinct primes $n$ "defines"?

Under Goldbach's conjecture, denote by $r_{0}(n)$ the smallest non negative integer $r$ such that both $n-r$ and $n+r$ are prime and by $k_{0}(n):=\pi(n+r_{0}(n))-\pi(n-r_{0}(n))$, so that $k_{0}(n)$ ...
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### Can a smooth manifold be realised as the image of a smooth function?

Consider, $M$, a smooth $m$ dimensional submanifold of $\mathbf R^n$. Does there exist a smooth map $X: \mathbf{R}^m\to\mathbf R^n$ such that $M=X(\mathbf R^m)$? $X$ may have points at which the ...
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### Around similar inequalities than an inequality due to Nicolas, that involve products of consecutive Ramanujan primes

This is cross-posted (and this post is a version to ask just around the veracity of Conjecture 1) as the post with identifier 3594907 and same title), that I've edited on Mathematics Stack Exchange ...
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### Projective scheme over the integers

Let $X$ be a projective scheme over $Spec(\mathbb{Z})$. Let $X_{p}$ be the reduction at $p$ of $X$. If for any prime $p$, $X_{p}$ is normal, can we deduce $X$ is normal? Or any counterexamples?
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### Solution to non-autonomous delay differential equation?

If you define a special function called the Lambert W function, you can explicitly solve the classic delay differential equation $x'(t) = x(t - a)$ by supposing the solution is some $\exp(\lambda t)$ ...
1 vote
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### Diffeomorphism induced by small perturbation

Consider the surface $S_{\epsilon}$ defined as: \begin{align} %S&=\{\vec x \in \mathbf{R}^3: x=0\}, \\ S_{\epsilon}&=\{\vec x \in \mathbf{R}^3:\epsilon (x^2 + y^2 + z^2 - 1) + x=0\}. \end{...
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### Searching for cofinal subsets of directed sets subject to finite constraints

Let $(P,\leq)$ be a directed set with uncountable cofinality. For every element $p\in P$, we are given a finite set $c_p\subset P\smallsetminus \{p\}$ of "incompatible elements". We say that ...
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### Counting the number of free bases of $F_n$ with elements of bounded length

Let $F$ be a free group of finite rank and fix a free generating set $X$ of $F$. Let $P_r$ denote the set of all free generating sets of $F$ whose elements have length bounded by $r$ (when considered ...
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Lemma. Let $E$ be a rank $r$ nef vector bundle over a polarized smooth complex projective variety $(X,H)$ of dimension $n\leq r$. Then for any $t\in\mathbb{R}_{\geq0}$: $$\sum_{k=0}^nt^{n-k}\int_Xc_k(... 4 votes 0 answers 111 views ### Surface with \Omega_X globally generated and singular Albanese image This question is inspired by abx's comment to my previous question MO430933. Let X be a complex surface of general type, and denote by$$a \colon X \to \operatorname{Alb}(X)$$the Albanese map of X... • 62.5k -3 votes 0 answers 30 views ### How to decide a threshold so that I can make small off-diagonal values of a diagonally dominant matrix to zero [closed] I have a matrix C which is diagonally dominant. Most of the off diagonal values of C are of the order of 1e-15 or smaller. I need to replace these small values by zeros with some limiting threshold. ... 1 vote 0 answers 23 views ### Associating a matroid to a uniform hypergraph For a fixed ground set [n]=\{1,\ldots,n\}, and for any matroid M on [n], specified as a collection of bases B_M, the corresponding matroid basis polytope P_M is defined to be the convex hull ... 0 votes 0 answers 67 views ### find all q such p\mid\left(\left(\dfrac{p-1}{q}\right)!\right)^q+1？ letp be prime number, following is well known$$p|\left(\left(\dfrac{p-1}{2}\right)!\right)^2+1$$proof:link1 and this post have prove link2$$p\nmid \left(\left(\dfrac{p-1}{3}\right)!\right)^3+1$$... • 3,644 4 votes 1 answer 71 views ### Extending a metric in a bi-Lipschitz way Suppose we are in the following situation: (X,d) is a metric space and Y is a subspace of X. Furthermore we have a different metric \delta defined on Y such that \delta is bi Lipschitz ... 1 vote 0 answers 22 views ### Non-vanishing of generalized minors on T-stable unipotent subgroups Let G be a complex simply connected algebraic group, T a maximal torus of G and B, B^- Borel subgroups which are opposite with respect to T and let U (resp. U^-) be the unipotent ... 5 votes 1 answer 250 views ### Day and Lack's "Limits of small functors": Lemma 2.3 I've been trying to understand the (4 line!) proof of Lemma 2.3 of Limits of small functors, on small functors into copresheaf categories \mathbf{Set}^\mathcal C. To me it seems to be using that the ... 5 votes 0 answers 123 views ### Is it true that the \mathbb{F}_p-rank of a linear combination of matrices is usually not smaller than its \mathbb{Q}-rank? Consider fixed 3 \times 3 integer matrices A_1,A_2,A_3 and the \sim H^3 linear combination matrices A(\mathbf{h})=h_1A_1+h_2A_2+h_3A_3 where h_1,h_2,h_3 are integers with \vert h_i\vert \le ... 2 votes 1 answer 68 views ### Pair of laminations that fill on a closed surface Let S be a hyperbolic surface of genus g \geq 2. A discrete geodesic lamination on S is a set of disjoint, simple, closed geodesics. Let L_{1} and L_{2} be two discrete geodesic laminations ... • 23 0 votes 0 answers 39 views ### A decision problem of an inverse problem in finite group theory A finite group G is called integral if there is a finite group H such that G\cong H'. In Araujo, Cameron, Casolo, Matucci's paper, integrals of groups, they tried to solve a problem as following:... 0 votes 0 answers 26 views ### Eigenvalues of orthogonal group element Let q be a quadratic form over a nonarchimedean local field F, and let \operatorname{O}(q) be the corresponding orthogonal group. Let g\in\operatorname{O}(q) be semisimple. Can we know ... • 551 0 votes 0 answers 30 views ### Intersection of certain subsets in a split connected reductive group G related to affine open cover of G/B Let k be a field of characteristic zero and G a split connected reductive group over k. Moreover, let T be a split maximal torus of G and B\supset T a Borel subgroup. Additionally, we ... • 911 2 votes 0 answers 44 views ### Explicit estimates on summability kernels A "summability kernel" is a sequence of functions k_n:[0,1)\to \mathbb C such that$$ \int_0^1 k_n(t) \mathrm d t =1, \int_0^1 |k_n(t)| \mathrm d t =O(1),$$with an implied constant ... • 2,675 0 votes 0 answers 116 views ### Relation between 3-term Plücker relations and more than 3-term Plücker relations \DeclareMathOperator\Gr{Gr}Let \Gr(k,n) be the Grassmannian variety of k-planes in an n-dimensional vector space. The coordinate algebra \mathbb{C}[\Gr(k,n)] is generated by Plücker ... • 5,885 0 votes 0 answers 51 views ### Is the element in the connected component? I posted this question at stack exchange, got two upvotes but no answer. If it doesn't belong here, please let me know. In the algebraic group G = PGL_{8}(\mathbb{C}), there are two involutions ... 3 votes 1 answer 299 views ### Trigonometric Diophantine equation Is there a general method to solve the equation P(x_1,x_2,...,x_n)=0 with P is a polynomial in n variables with integer coefficients and x_k=\cos(q_k\pi) with q_k is a rational number? This ... • 1,040 -3 votes 0 answers 45 views ### Is there a closed form of the net present value formula? [closed] The net present value formula is as follows:$$ npv(q, n, r)=\sum_{i=0}^{n}q\left(1-r\right)^i $$where n is the number of periods, q is the payment in each period, and r is future discount ... • 95 0 votes 0 answers 40 views ### How to prove this inequality for the norm  \|\cdot\|_{1,\infty} ? Let  \{a_k\}  is a positive sequence. For  0<p<\infty , space  L^{p,\infty}  is defined by$$ \left\{f:\|f\|_{p,\infty}=\inf\left\{C>0:a_f(\lambda)\leq C/\lambda^p\right\}\right\} $$... 0 votes 1 answer 49 views ### \omega-homogenous space which is not \omega_1-homogenous Consider a metric space (X,d) and let \kappa be a cardinal. We say that (X,d) is \kappa-homogenous, if every (surjective) isometry h:X_1 \to X_2 between subspaces of (X,d) of size < \... 4 votes 1 answer 75 views ### Is there any example of a Lindelöf space that has no Menger dense subspaces? A space X is said to be Menger if for each sequence (\mathcal{U}_n) of open covers of X, there is a sequence (\mathcal{V}_n) such that \mathcal{V}_n is a finite subcollection of \mathcal{U}... • 81 1 vote 0 answers 151 views +200 ### Hodge theory of the AJS category (proof of Lusztig's conjectures in positive characteristic) Recently, in this survey paper (https://arxiv.org/abs/1212.0791) Elias-Williamson describe a Hodge theoretic approach to the proof of Kazhdan-Lusztig conjectures; it is essentially equivalent to the ... • 1,748 7 votes 1 answer 256 views ### Does the limit of x_n, defined by x_{n+1}=1/(m+1-nx_n) exist? Let m be positive integer, and consider the recursion$$x_{n+1}=\frac{1}{m+1-nx_n}.$$Does the limit of x_n exist? I'm guessing the limit doesn't exists for any m. • 3,644 4 votes 0 answers 139 views +50 ### \Sigma_*-product is not \sigma-countably compact In Arhangel'skii's book "Topological function spaces" there is a part where the author uses that, if \kappa>\omega is a cardinal number, then the space$$\Sigma_*(\kappa):=\left\{x\in \...
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The following statement is from the article Compact Coadjoint Orbits by John Rawnsley: If $\mathcal{O}$ is a compact coadjoint orbit for the group $G$ then there is a closed normal subgroup $H$ of $G$...
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### Square-root lattices: where do they appear?

As an experimental physicist working on crystallography I'm often dealing with the reconstruction of an object from intensity data that emerge from an imaging device. In mathematics the problem is ...
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