0
votes
0answers
4 views

Girsanov theorem with Geometric Brownian Motion

I am not a student in mathematics, but I am trying to use the following Theorem 8.6.6 (Girsanov theorem II) of Oksendal's SDE with geometric Brownian motion $S_{t}$ instead of the standard Brownian ...
2
votes
0answers
10 views

Three old questions on the Sacks forcing

I came across the two following Qs in 1970. Find reals $a,b$ such that $a$ is Sacks over $L[b]$ and vice versa $b$ is Sacks over $L[a]$. Note that a Sacks $\times$ Sacks generic pair definitely does ...
0
votes
0answers
9 views

Categories with binary relations as objects

For the category of functions, pairs of functions making commutative diagrams are the canonical morphisms $\alpha:f\rightarrow g$. For binary relations there is an alternative, to consider the ...
0
votes
0answers
25 views

A doubt on balaji meyn's ergodic theorem paper

I have a question regarding the classic paper by Balaji and Meyn: "Multiplicative ergodicity and Large Deviations for an Irreducible Markov Chain". Consider a recurrent apeiodic irreducible Markov ...
-2
votes
0answers
27 views

Can the Einstein Field Equations be written as Difference Equations?

Does anyone know if the Einstein Field equations have ever been written as Difference Equations, and if so does that simplify anything or produce solutions not available in the usual Differential ...
1
vote
0answers
17 views

Helmholtz equation Poynting vector integral

The Maxwell's equation for harmonic time dependent field in vacuum is \begin{align} \nabla \times B + i\omega E &= 0\\ \nabla \times E - i\omega B &= 0 \\ \nabla \cdot B &= 0 \\ \nabla ...
0
votes
0answers
60 views

Notation for “partial” derivative [on hold]

Say I have a function like: $f(x,c,y) = x+c+y$ say that $c$ is a function of $x$, $c(x)$ (i.e. $x$ and $y$ are the only truly independent variables). How would I write the notation for the ...
3
votes
0answers
59 views

minimal energy of affine Lie algebra reps

Let $\mathfrak g$ be a simple Lie algebra. Let $\widetilde{L\mathfrak g}$ be the universal central extension of $L\mathfrak g:=\mathfrak g[t,t^{-1}]$. Let $V_\lambda$ be a positive energy ...
1
vote
0answers
39 views

Submodule embeddable in a finitely generated module

I have a terminology question. For a commutative ring $A$ (not necessarily Noetherian), $A$-modules that are isomorphic to an $A$-submodule of a finitely generated $A$-module form a fairly good class ...
2
votes
0answers
84 views

If $B\subseteq A$ are free & finite rank $R$-algebras, is $R\to A \otimes_B R$ injective?

(In this question, all rings and algebras are commutative with identity.) I have a situation that boils down to the following data: a ring $R$, an $R$-algebra $A$ with a subalgebra $B$ such that $A$ ...
4
votes
0answers
84 views

Invariant obstructions to gluing Galois representations on elliptic curves

Let $E$ be an elliptic curve over $\overline{\mathbb F_p}$, or another separably closed field of characteristic $p$. Let $K$ be the function field of $E$, and let $K_p$ be the local field at a point. ...
-4
votes
0answers
47 views

How do you generate orthogonal Rotations in d-dimensions? [on hold]

I know that matrices that satisfy $\mathbf{x^TMx} > 0$ for all $\mathbf{x} \in \mathbb{R}^d$ are called positive definite matrices. It seems to me that a matrix that represents an $90^{\circ}$ ...
1
vote
1answer
100 views

Realizing algebraic curves as complete intersections

I have two related questions about smooth complete algebraic curves (over $\mathbb{C}$). Does there exist a smooth complete algebraic curve $X$ that cannot be embedded as a complete intersection in ...
3
votes
0answers
60 views

reference on aperiodicity and cluster

From this image: I believe there is a message relating those clusters drawn in picture and aperiodic tiling. Does anyone have some reference on this? Thank you :)
2
votes
1answer
114 views

Proof of the Dunford-Pettis theorem

I would like to know where to find a complete proof of the Dunford-Pettis theorem: A sequence $(f_n)_{n\geq 0} \subset L^1$ is uniformly integrable if and only if it is relatively compact for the weak ...
2
votes
2answers
80 views

Software producing complex trees

Does anyone know any kind of graph software that could produce graphs like this for publication? Those links and crosses and numbers actually needs to be presented…. Thank you:) One small update, ...
0
votes
0answers
67 views

How to find generators to Mordell weil groups of elliptic curves using Sage software? [on hold]

I am new to sage.I want to find generators to Mordell Weil group of the Elliptic Curve $y^2=x^3-6321363052$. While using sage it gives error as increase decent and then make use of same command but I ...
-1
votes
0answers
54 views

Complex conjugation of positive roots [migrated]

I have a simple question about root systems. Suppose that $G$ is a connected reductive group over the reals $\mathbb{R}$, and $T\subset G$ is a maximal torus (by this I mean that $T_{\mathbb{C}}$ is a ...
0
votes
0answers
33 views

Behavior of the integral of products of probability densities

Assume $z \in \mathbb{R}^m$ and $x \in \mathbb{R}^n$. Assume we have proper density function $P(z)$ and proper conditional density function $P(x|z)$. We give the definition $$ T(x_1,\ldots,x_n) := ...
6
votes
1answer
374 views

Why is the section conjecture important?

As in the title, I want to know the reason for importance of the section conjecture. Of course, the statement of conjecture is important as itself, even I cannot fully grasp the soul of it. However, ...
0
votes
0answers
33 views

Homotopy addition theorem and lifting in a certain diagram

If I am confusing somewhere here, please excuse me and ask me to clarify. The proof I am having trouble with is lemma 1.19 of Goerss-Jardine, pg. 398. I have tried to isolate the troublesome ...
9
votes
1answer
216 views

Erdös cardinals and ineffable cardinals

In Cantor's Attic it is stated that an $\omega$-Erdös cardinal is a stationary limit of ineffable cardinals, and Jech book is given as a reference, but I cannot find this result in that book. I have ...
7
votes
1answer
126 views

Trapping a particle

A particle starts a brownian walk in the middle of a long tunnel in the plane, at one end of the tunnel is a circular region X, at the other end a region Y of given area A. Does the shape of region ...
0
votes
1answer
59 views

reference request for automata of this type [on hold]

Consider the list of length $m$ $(1,0,\dots 0)$ we call this list $l_1$, we now define a sequence of lists recursively, where $l_1$ is the previous list, and if $l_n$ is the list $(a_1,a_2\dots a_n)$ ...
1
vote
1answer
71 views

Strong maximum principle for weak solutions

Suppose I have a linear parabolic equation with solutions in the Bochner-Sobolev spaces (eg. $L^2(0,T;H^1) \cap H^1(0,T;H^{-1})$). Is it possible to obtain a strong maximum principle with a proof that ...
3
votes
0answers
45 views

Strong convexity of the trace of the square root of a matrix function

Any clues about how to prove that the following function is strongly-concave in $x$? (We conjecture it is $2$-strongly concave but cannot prove it. We have already proved strict concavity through ...
-3
votes
0answers
29 views

Finding Distance with Missing Coordinate Set [on hold]

I have a line that runs through the orgin with a slope of 2, with a distance of 5 from the orgin what are the coordinates and how did you solve? I did some searching online and only found lectures on ...
0
votes
0answers
64 views

Relation between long exact sequences and Derived functors [on hold]

I know that if i have a short exact sequence of chain complexes $$0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$$ then i can extend it to long exact sequence of homology groups as ...
-1
votes
0answers
24 views

impossibility and mode of convergence [on hold]

I have $\mathrm{Pr}(a>b)=0$ with $b\in[0,\infty]$, then can i have $a<0$? If this is true, which type of convergence is this?
0
votes
0answers
90 views

Any result, example or conjecture about the computational complexity of transcendental number that can not be computed by linear time Turing Machine [on hold]

The Hartmanis-Stearns Conjecture is restated by Prof.Lipton as: Suppose that a linear time Turing Machine computes the first $n$ digits of the real number $r$ in base ten. Then, the number is either a ...
11
votes
2answers
429 views

Are the quaternions not uncountably categorical?

Boris Zilber has argued that the field of the complex numbers is "logically perfect". For one thing, the theory of an algebraically closed field of characteristic zero is uncountably categorical: it ...
13
votes
2answers
261 views

Probing the generalization of the abc conjecture to more than 3 variables

Browkin and Brzezinski, in "Some remarks on the $abc$-conjecture", Math. Comp. 62 (1994), no. 206, 931–939, state the following generalization of the $abc$ conjecture to more than three variables: ...
0
votes
0answers
66 views

Are there any relevance between coefficients of simple continued fraction of quadradic algebraic number and algebraic number with degree $2^n$ [on hold]

Let $\sqrt{c}$ be quadratic algebraic number, We know that $[a_0;a_1,a_2,\dots ]$ the coefficients of simple continued fraction of $\sqrt{c}$ the quadratic algebraic number is periodical. ...
6
votes
1answer
155 views

Adams e-invariant

In "On the Groups J(X) - IV", Adams introduces the $e$-invariant, which turns out to be closely connected to the image of the $J$-homomorphism, but he introduces it in a more general setting. He has ...
9
votes
1answer
94 views

Completely positive maps-equivalent definition

The most usual definition of the completely positive map (c.p.) between two C*-algebras (say, unital) is the following: $\sigma: A \to B$ should satisfy $\sigma(1)=1$ and for each $n \in \mathbb{N}$ ...
-3
votes
0answers
104 views

Numbers half way between two primes [on hold]

Is every integer greater than 3 half way between two primes?
1
vote
0answers
114 views

varieties whose canonical bundle has finite order in Pic?

Is there a structure theorem for such varieties? If X is a smooth and proper/projective variety whose canonical bundle $\omega_X$ has finite order in the Picard group, do we know anything about X? ...
9
votes
0answers
118 views

Cubic fields correspond to $3$-torsion ideals in quadratic fields, or to order $3$ characters of quadratic class groups?

I was watching Dick Gross's laudation for Manjul Bhargava, followed up by one of Bhargava's talks, and I realized I was confused about something. Bhargava says (around 21 minutes) that the orbits of ...
-4
votes
0answers
34 views

Adjoint quotient in terms of a Chevalley basis [on hold]

Has anyone put the adjoint quotient (of a Lie algebra) in terms of a Chevalley basis? If so, do you have a reference?
1
vote
0answers
170 views

Quotes from Connes

I found the following remark by Connes HERE: "the main quality of the homotopy type of a manifold is to satisfy Poincare duality not only in ordinary homology but in K-homology with the Fredholm ...
7
votes
0answers
116 views

Can Suslin lines ever be orderings of abelian groups?

I am interested in realizing linear orders as orderings of abelian groups. In particular, can Suslin lines (and other classes of line) be realised in this way? Let $\mathcal{C}$ be a class of ...
9
votes
2answers
329 views

Elliptic curves with trace of Frobenius values always congruent to 0 modulo 2

Let $E/\mathbb{Q}$ be an elliptic curve and suppose that the trace of Frobenius values are such that $a_{p}(E) \equiv 0 \pmod{2}$ for all odd primes avoiding the conductor. Is it the case that $E$ ...
8
votes
1answer
162 views

Maximum length of a chain of topologies on $\Bbb R$

Let $\frak T$ be a totally ordered set of topologies on $\Bbb R$. Is $|\frak T|\le |\Bbb R|$?
-4
votes
1answer
179 views

When are two algorithms essentially the same?

Inspired by Blass/Dershowitz/Gurevich's paper When are two algorithms the same? (which was referenced in another context here) I tried to boil down the question to the following situation: Consider ...
16
votes
2answers
502 views

construction of nonmeasurable sets

I have a history question for which I've had trouble finding a good answer. The common story about nonmeasurable sets is that Vitali showed that one existed using the Axiom of Choice, and Lebesgue et ...
0
votes
0answers
42 views

Conditional probabilities in epidemic model

I was contemplating an epidemic model where infection and recovery rates are determined by links. Here node $i$ is infected first and recovers at a rate $\mu_i$. For all other nodes, the recovery is ...
0
votes
1answer
102 views

Dimension of Commutator Space

For each $n\times n$ matrix $A$ with real entries the set $$C(A)=\{X\in M_n(\mathbb{R}): AX=XA\}$$ is obviously a linear subspace of $M_n(\mathbb{R})$. Can we recognize the dimension of this ...
2
votes
0answers
39 views

Image of the typenorm contains the squares

I am having a look at the paper Explicit CM-theory for level 2-structures on abelian surfaces by Bröker, Gruenewald and Lauter, and there is an argument which I don't understand. The main reason being ...
-2
votes
0answers
69 views

Is there an example of an SDE that has no weak solution? [on hold]

So far I couldn't find an example for an SDE for which there exists no weak solution. Do you know one?
5
votes
1answer
114 views

Tensor product of certain Sobolev spaces on non-compact manifolds

Let $M$ be a non-compact Riemannian manifold of bounded geometry (i.e., its injectivity radius is uniformly positive and the curvature tensor and all its covariant derivatives are bounded in ...

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