neuroscience - If D1 receptors stimulate adenylate cyclase (through GPCRs) and D2 receptors inhibit it, then why do mutations in both have similar effects?

D1 and D2 both refer to specific types of dopamine receptors.

I'm sure it has something to do with the fact that the D1 receptors are in regions different from D2 receptors.

I know that adenylate cyclase usually triggers a signal transduction cascade that leads to increased cAMP+calmodulin, resulting in increased gene expression of proteins that help promote long-term potentiation of postsynaptic neurons (presumably lowering the absolute value of the voltage threshold that's needed to trigger another action potential, which increases neuron excitability).

But an increase in dopamine is going to cause more neuron excitability in postsynaptic neurons in some regions (those that have D1 receptors), and less in others (those that have D2 receptors). What then explains the fact that increased expression at all dopamine receptors can help modulate attention span?

I know that D3 and D4 are also involved, but they have to be either excitatory, inhibitory, or neutral.

linear algebra - Closedness of finite-dimensional subspaces

This holds indeed for complete fields: see Theorem 2, Section I.2.3, of Bourbaki's "Espaces Vectoriels Topologiques".

Here is the argument.

Let $K$ be a (not necessarily commutative) field equipped with a complete nontrivial absolute value $xmapsto|x|$, let $n$ be a positive integer, let $tau$ be a Hausdorff vector space topology on $K^n$, and let $pi$ be the product topology on $K^n$.

THEOREM $tau=pi$.

REMINDER A topological group $G$ is Hausdorff iff {1} is closed. [Proof: {1} is closed $Rightarrow$ the diagonal of $Gtimes G$ is closed (because it's the inverse image of {1} under $(x,y)mapsto xy^{-1}$) $Rightarrow$ $G$ is Hausdorff.]

LEMMA The Theorem holds for $n=1$.

The Lemma implies the Theorem. We argue by induction on $n$. The continuity of the identity from $K^n_pi$ to $K^n_tau$ (obvious notation) is clear (and doesn't use the Lemma). To prove the continuity of the identity from $K^n_tau$ to $K^n_pi$, it suffices to prove the continuity of an arbitrary nonzero linear form $f$ from $K^n_tau$ to $K_pi$. By induction hypothesis, the kernel of $f$ is closed, and the Theorem follows from the Reminder and the Lemma.

Proof of the Lemma. We'll use several times the fact that $K^times$ contains elements of arbitrary large and arbitrary small absolute value. As already observed, we have $tausubsetpi$. If $x$ is in $K^times$, write $B_x$ for the open ball of radius $|x|$ and center 0 in $K$. Let $a$ be in $K^times$, and let $tau_0$ be the set of those $U$ such that $0in Uintau$.

It suffices to check that $B_a$ contains some $U$ in $tau_0$.

We can find a $b$ in $K^times$ and a $V$ in $tau_0$ such that $a$ is not in $B_bV$, and then a $c$ in $K$ with $|c|>1$ and a $W$ in $tau_0$ such that $a$ is not in $B_cW$. Then $U:=c^{-1}W$ does the job.

gr.group theory - Extensions isomorphic as groups but not congruent or pseudo-congruent

I'm looking for an example of a finite abelian group A and a finite group G acting trivially on A such that there are two extensions $E_1$ and $E_2$ with base A and quotient G (i.e., they are both central extensions, and hence both give corresponding elements of $H^2(G,A)$) and:

1. $E_1$ and $E_2$ are isomorphic as abstract groups.


2. Under the natural action of $operatorname{Aut}(G) times operatorname{Aut}(A)$ on $H^2(G,A)$ (by pre- and post-composition with 2-cocycles that then descends to action on cohomology classes), the cohomology classes corresponding to $E_1$ and $E_2$ are not in the same orbit.

Basically condition (2) states that $E_1$ and $E_2$ are not only not congruent extensions, they are not even congruent up to a relabeling of the subgroup A and the quotient G. Another way of putting this is that there is no isomorphism between $E_1$ and $E_2$ that sends the A inside $E_1$ to the A inside $E_2$.

The analogous statement with a nontrivial action of G on A is also of interest to me. In this latter case, though, the entire group $operatorname{Aut}(G) times operatorname{Aut}(A)$ does not act.

I think that examples exist (because of my experience with finding examples for similar specifications) but there may well be a proof to the contrary.

lie groups - Is the space of volume-preserving maps path-connected?

This is a clarification of another post of mine.

Fix $n$ a positive integer. Let $SL(n)$ have its usual matrix representation, so that it really is the codimension-one subset of $M(n) = mathbb R^{n^2}$ cut out by the degree-$n$ condition that the determinant is $1$. So we have $n^2$ coordinate functions $A^i_j$ on $SL(n)$, $i,j = 1,dots,n$.

Let $U$ be a domain in $mathbb R^n$, with coordinates $x_1,dots,x_n$. Consider the set $mathcal S$ of smooth functions $f: U to SL(n)$ satisfying the differential equation $frac{partial f^i_j}{partial x^k} = frac{partial f^i_k}{partial x^j}$ for each $i,j,k = 1,dots,n$ (of course, $f^i_j = A^i_j circ f$ is the $(i,j)$th coordinate of $f$).

(Why would you care about $mathcal S$? Because a smooth map $g: U to mathbb R^n$ is volume-preserving if and only if $frac{partial g^i}{partial x^j} in mathcal S$, and every element of $mathcal S$ arises this way; indeed, $mathcal S$ is the space of volume-preserving maps up to translations.)

Let's agree that a smooth path in $mathcal S$ is a smooth function $F: [0,1] times U to SL(n)$ such that for each $tin [0,1]$, $F(t,-) in mathcal S$.

Question: Is $mathcal S$ smooth-path-connected? I.e. given $f_0, f_1 in mathcal S$, does there exist a smooth path $F$ so that $F(0,-) = f_0$ and $F(1,-) = f_1$?

If the answer is "no" in general, is it "yes" for sufficiently nice domains $U$ (contractible, say, or with compact closure and require that each $fin mathcal S$ extend smoothly to a neighborhood of the closure, or...)?

noncommutative geometry - Finding the Universal Ideal of a (Covariant) Differential Calculus

I don't know if you still care, but I think I found the answer to your question.

Look at Proposition 1 in Chapter 14 of Quantum Groups and Their Representations by Klimyk and Schmudgen. It shows that there is a bijection between left-covariant first-order differential calculi over a Hopf algebra $H$ and right ideals of the kernel of the counit of $H$, and it shows how the relations in a first-order calculus are obtained from the ideal. I have never worked with these things, so I don't know how tractable the computations are, but as far as I can tell from quickly scanning it seems that all the maps are at least explicitly defined.

As for higher order calculi, I am not sure whether/how these results extend. But at least maybe this is a good start?

ag.algebraic geometry - Relation between l-adic and l'-adic geometric monodromy

Suppose $X$ is a smooth family of algebraic varieties over the base $B:=mathbb{P}^1backslashlbrace0,1,inftyrbrace$ over $overline{mathbb{Q}}$; then we can form the relative $l$-adic cohomology on the base, which will be (having fixed some cohomological degree of interest) an $l$-adic sheaf; we can think of it as a $mathbb{Q}_l$ vector space $V_l$ equipped with a $pi_1(B)$ action. (Algebraic $pi_1$, of course.)

We can do the same with $l'$-adic cohomology for some other prime $l'$, getting a vector space $V_{l'}$ over $mathbb{Q}_{l'}$ equipped with a $pi_1(B)$ action.

As far as I can see, there ought to be a relatively tight connection between these two groups. For instance, it seems morally like we ought to be able to choose pro-generators $alpha,beta$ for $pi_1(B)$, and find some matrices $M_alpha,M_beta$ over $overline{mathbb{Q}}$, such that, w.r.t suitable bases of $V_l,V_{l'}$, $alpha$ acts via $M_alpha$ on both $V_l$ and $V_{l'}$ (thinking of $M_alpha$ as determining an $l$- and $l'$- adic matrix in turn) and similarly for $M_beta$. Can this indeed be done? If not, can you do something close? If it can, what's a good reference for a precisely stated theorem---rather than just intuition---for these things?

dg.differential geometry - Following curves on S^n

No, your solutions do not need to be closed curves. For example, take the vector field "multiplication by $i$" on the $3$-sphere, thought of as the sphere of unit quaternions. That has closed solutions (Hopf fibration fibres). With a little bump function you can perturb this vector field to ensure there's only two closed solutions -- make the perturbation "away" from one closed solution and "towards" the other. This can be made concrete in many ways but I hope you get the idea.

edit: If you're interested in minimizing the number of closed orbits you can readily get down to one, on $S^3$. The meta-idea above is that the normal bundle to the Hopf fibration is the pull-back of the tangent bundle to $S^2$. $S^2$ has a flow with only one fixed point, so pull that vector field back to the normal bundle of the Hopf fibration, add it to "multiplication by $i$" and now you have a vector field with only one closed orbit (the Hopf fibre over the fixed point to your vector field on $S^2$).

co.combinatorics - Disjoint stable sets in tournaments

Let $(V,A)$ be a tournament. A subset of vertices $V'subseteq V$ is stable if
there exists no $vin Vsetminus V'$ such that $V'cup${$v$} contains an inclusion-maximal transitive subtournament with source $v$.

(In other words, $V'$ is stable if for every transitive subtournament $Tsubseteq V'cup${$v$} with $vin T$ and $(v,x)in A$ for all $xin Tsetminus${$v$}, there is a $win V'$ such that $(w, x)in A$ for all $xin T$.)

Is it true that no tournament contains two disjoint stable sets?

The statement would imply that every tournament contains a unique minimal stable set, which would have several appealing consequences in the social sciences. The statement is a weak version of a conjecture by Schwartz (see this paper and the references therein). Computer analysis has shown that there exists no counter-example with less than 13 vertices.

gr.group theory - direct limit of free complemented subgroups

Edit: The answer below has been modified to reflect the comments.

My guess is that $G$ is forced to be projective, hence free, in this situation. To show this, we need to verify that $mathrm{Hom}(G,-)$ is an exact functor. As we have an identification of functors $mathrm{Hom}(G,-) simeq lim_i mathrm{Hom}(F_i,-)$, applying $mathrm{Hom}(G,-)$ to an exact sequence

$0 to A to B to C to 0$

of abelian groups, we get an induced sequence

$0 to lim_i mathrm{Hom}(F_i,A) to lim_i mathrm{Hom}(F_i,B) to lim_i mathrm{Hom}(F_i,C) to R^1 lim_i mathrm{Hom}(F_i,A) to ...$

So it suffices to show that $R^1 lim_i mathrm{Hom}(F_i,A)$ vanishes for any abelian group $A$. Is this true?

Here's a not-so-well-thought-out idea: if I chased elements correctly, an affirmative answer to the question above follows from the bijectivity of the natural map $mathrm{Hom}(F_j,A) to lim_{i < j} mathrm{Hom}(F_i,A)$, for j sufficiently big. After making the harmless assumption that the system $(F_i)$ consists of all finitely generated saturated subgroups of $G$, the preceding bijectivity question translates to: given a free abelian group F of finite rank, when is the natural map $mathrm{colim}_i H_i to F$ an isomorphism, where the indexing set I is the poset of all proper saturated subgroups $H_i subset F$. I think the answer to this question is yes when the rank of $F$ is at least $3$ (which is enough for the application at hand), but I'm not sure.

dg.differential geometry - Lipschitz equivalence of Riemannian metrics

As Dmitri says any two Riemannian metrics on a compact manifold are Lipschitz equivalent. The proof is quite simple.

Consider $g$ and $h$ two metrics on $M$, Let $UM$ be the unit tangent bundle, since $M$ is compact, $UM$ is compact. Then you see that $f:UMto mathbb{R}$ defined by $f(x)=frac{g(x,x)}{h(x,x)}$ is continuous and strictly positive. By compactness, it is bounded above and below by positive constants.

What is the criticality of the ribosome binding site relative to the start codon in prokaryotic translation?

I found an oldish paper on this topic (from 1994). Here's a summary:

Determination of the optimal aligned spacing between the Shine-Dalgarno sequence and the translation initiation codon of Escherichia coli mRNAs. by Chen, Bjerknes, Kumar, & Jay. Nucleic Acids Research. (1994)

Experiment

The authors constructed a series of synthetic RBS regions that varied the length separating a synthetic 5-nt Shine-Dalgarno sequence from the start codon. The regions varied in size between 2 to 17 nt. They assayed the activity of a downstream enzyme, chloramphenicol acetyltransferase.

Conclusion

The authors concluded that the optimal spacing between a consensus 5-nt Shine-Dalgarno sequence (5'-GAGGT-3') and the start site was 5 nt. Note: this synthetic SD was made of the last 5 nt of a 9 nt SD consensus sequence. They also tested a synthetic SD made from the first 5 nt (5'-TAAGG-3')of the consensus SD. In this case they found the optimal distance was 9 nt.

So the optimal distance depends on where your desired SD aligns with the consensus SD sequence, which optimally is 5 nt from the start. Read on for more details.

Details

• the RBS is considered to be large, extending 20bp on either side of a core Shine-Dalgarno (SD) sequence. These days, I often hear of the RBS spoken of in sizes that are equivalent to the SD. So in the parlance of the paper, you question is rephrased as "how does distance of the SD from the start codon effect translation?"




• the canonical SD sequence referenced in the paper is 5'-UAAGGAGGU-3'. It is 9 nucleotides long. Distances between the SD and the start codon are defined as the number of nucleotides separate the 3' Uracil of the SD from the Adenine of the start AUG.




• Example: the distance is 5 nt in the following mRNA

  
  
  

         5'.....UAAGGAGGUnnnnnAUG......3'
  




• if the SD is not a complete 9 nt long, the distance is between the position of where the canonical Uracil would occur. In the following example, the distance is still calculated as 5 nt:

  
  
  

         5'.....UAAGGAnnnnnnnnAUG......3'
  




• the average distance between the SD sequence and the start codon varies considerably and on average is 7 nt. Other investigators have found "optimal" spacing (circa 1994) ranged from 5 to 13 nt.




• the SD site complements with region on the 16s rRNA. The start codon complements to the anticodon of fMet-tRNA loaded into the ribosomal P-site. So there are two distinct sites on the ribosome that contact the mRNA during translation initiation.

Also good to read

Note

I enjoyed researching this question because I found my own knowledge to be lacking solid empirical details. I do not have any direct experience besides this little lit review with this topic.

physiology - Why is coffee a laxative?

Coffee does have an effect on the peristaltic movement in the bowel.

Coffee increases rectosigmoid motor activity within 4 min after ingestion in some people. Its effects on the colon are found to be comparable to those of a 1000 kCal meal. Since coffee contains no calories, and its effects on the gastrointestinal tract cannot be ascribed to its volume load, acidity or osmolality, it must have pharmacological effects. Caffeine cannot solely account for these gastrointestinal effects.

Effectively, decaf and regular coffee stimulate peristaltic movement in the colon as effectively as a meal does. Caffeine is not the active agent then, but some other compound in coffee.

Source: Boekema PJ, Samsom M, et al. Coffee and gastrointestinal function: facts and fiction. A review. Scand J Gastroenterol Suppl. 1999;230:35-9. PMID 10499460.

Depth Zero Ideals in the Homogenized Weyl Algebra

• Let $mathcal{D}$ be the $n$th Weyl algebra $ mathcal{D} :=k[x_1,...,x_n,partial_1,...,partial_n] $, where $partial_ix_i-x_ipartial_i=1$.


• Let $widetilde{mathcal{D}}$ be its Rees algebra, which is $ mathcal{D} :=k[t, x_1,...,x_n,partial_1,...,partial_n] $, where $partial_ix_i-x_ipartial_i=t$, and $t$ is central.


• Let $mathcal{O}_X$ denote the polynomial algebra $k[x_1,...,x_n]$, which is a left $widetilde{mathcal{D}}$-module, where $t$ and all the $partial_i$ act by zero. NOTE: this is different than the homogenization of the standard $mathcal{D}$-module structure on $mathcal{O}_X$.

The question I am interested in is, how many generators does a left ideal $M$ in $widetilde{mathcal{D}}$ need before $Hom_{widetilde{mathcal{D}}}(mathcal{O}_X,widetilde{mathcal{D}}/M)$ can be non-zero? My conjecture is that $M$ needs at least $n+1$ generators. NOTE: Savvy Weyl algebra veterans will know every left ideal in $mathcal{D}$ can be generated by two elements; however, this is not true in $widetilde{mathcal{D}}$. There can be ideals generated by $n+1$ elements and no fewer.

The functor $Hom_{widetilde{mathcal{D}}}(mathcal{O}_X,-)$ acts as a relative analog of the more familiar functor $Hom_R(k,-)$ (where $R=k[t,y_1,...y_n]$). Therefore, the above question is analogous to asking "how many generators must an ideal $Isubseteq R$ have before $R/I$ can have depth zero?" The answer here is $n+1$, which follows from Thm 13.4, pg 98 of Matsumura (essentially a souped up version of the Hauptidealsatz).

In the noncommutative case, if you try to make this work with the $widetilde{mathcal{D}}$-module $k$ (where $t$, $x_i$ and $partial_i$ all act by zero), it doesn't work. The natural conjecture would be that $Hom_{widetilde{D}}(k,widetilde{mathcal{D}}/M)neq 0$ implies $M$ had at least $2n+1$ generators, except this fails even for the first Weyl algebra and $M=widetilde{mathcal{D}}x_1+widetilde{mathcal{D}}partial_1$ (since $widetilde{mathcal{D}}/M=k$).

However, it seems that things might work right for the relative module $O_X$, based on a fair amount of experimentation. It is easily true in the first Weyl algebra. Oh, and equivalent condition is to ask when $Hom_{overline{mathcal{D}}}(mathcal{O}_X,overline{M})neq 0 $, where $overline{mathcal{D}}=widetilde{mathcal{D}}/t$, and $overline{M}$ is $M/Mt$.

matrices - Random products of projections: bounds on convergence rate?

The von Neumann-Halperin [vN,H] theorem shows that iterating a fixed product of projection operators converges to the projector onto the intersection subspace of the individual projectors. A good bound on the rate of convergence using the concept of the Friedrichs number has recently been shown [BGM].

A generalization of this result due to Amemiya and Ando [AA] to the product of random sequences of projection operators drawn from a fixed set also shows convergence to the projector onto the intersection subspace.

My question is: are there any known bounds on the convergence rate for the latter problem analogous to the earlier one? In my application I'm only interested in the case of finite-dimensional Hilbert spaces.

[vN] J. von Neumann, Functional operators, Annals of Mathematics Studies No. 22, Princeton University Press (1950)

[H] I. Halperin, The product of projection operators, Acta. Sci. Math. (Szeged) 23 (1962), 96-99.

[BGM] C. Badea, S. Grivaux, and V. M¨uller. A generalization of the Friedrichs angle
and the method of alternating projections. Comptes Rendus Mathematique, 348(1–2):53–56, (2010).

[AA] I. Amemiya and T. Ando, Convergence of random products of contractions in Hilbert space, Acta. Sci. Math. (Szeged) 26 (1965), 239-244.

co.combinatorics - Making a non-monotone function monotone

Consider a function $f: {0,1}^n rightarrow {1..R}$. This function can be interpreted as a coloring $Color(v)$ of vertices in a unit n-dimensional hypercube in $R$ colors.

We say there is a directed edge $(v1, v2)$ in the hypercube if $v1$ and $v2$ differ in only one coordinate in n-dimensional space and this coordinate is equal to $0$ for $v1$ and to $1$ for $v2$.

Let's define $E(f)$ to be the number of non-monotone directed edges in this hypercube, i.e. edges $(v1, v2)$, such that $Color(v1) > Color(v2)$. Having a non-monotone function $f$ we want to make it monotone, changing its values in as few points of domain as possible. Let's denote $M(f)$ to be the minimal number of points where we need to change the values to make the function monotone.

There is a hypothesis that $M(f) le E(f)$ that I'm trying to prove.

Known results:

0) There exist such $f$ that $M(f) ge E(f)$ (if $E(f)$ is not too large, say $E(f) le 2^{n-1}$). This one is an easy exercise.

1) For $R=2$ the hypothesis is true. The method I know is rather difficult to describe briefly. If necessary, I can give a link (see EDIT).

2) For general $R$ it can be proved that $M(f) le E(f) log{R}$. The proof involves construction for $R=2$ and some range reduction technique.

I am interested in what techniques can be applied to prove or disprove this hypothesis. Any result better than $M(f) le E(f) log{R}$ (like $M(f) = O(E(f))$) will be interesting.

EDIT: Here is a link to M.Sc. thesis by Sofya Raskhodnikova, where results 1) and 2) can be found in Chapters 3 and 5 respectively.

EDIT: Here is a link to some informal description of motivation for this problem.

co.combinatorics - Red-blue alternating paths

I think even the stronger lemma is true where all paths will be edges:

Let $G$ be a simple graph, $V_k={v_1,dots,v_k}$ a subset of vertices and we have $deg(v_i)geq i$ for all $i=1,dots,k$. Then there is a matching which covers $V_k$.

PROOF:

If there is no edge between vertices from $V_k$ (so $V_k$ is stable) then we can consider only the edges between $V_k$ and $Vsetminus V_k$ (the induced bipartite graph between $V_k$ and $Vsetminus V_k$). The degree of the vertices from $V_k$ in this bipartite graph will be the same as in the original graph, so the condition still holds. From this it follows that $V_k$ satisfies the condition of Hall's marriage theorem thus there is a matching covering $V_k$. Indeed, let $Xsubseteq V_k$, and let $v_lin X$ be the vertex with the highest index in $X$. Then $|X|leq l$. On the other hand all the neighbors of $v_l$ are in the neighborhood of $X$ so $deg(v_l)leq |N(X)|$. Combining these two observation with our condition of $deg(v_l)geq l$ we get that $|X|leq |N(X)|$

If there is an edge $(v_iv_j)$, where $v_i,v_jin V_k$, then let's choose such an edge where the sum of indexes $i+j$ is the smallest possible and suppose that $i<j$. Let's delete $v_i$ and $v_j$ from the graph. All the other vertices from $V_k$ can have their degree decrease with at most 2. If the degree of $v_l$ decreases by 1 then $v_l$ is a neighbor of $v_i$ or $v_j$ and because of $i+j$ being minimal we have $i<l$. If the degree of $v_l$ decreases by 2 then $v_l$ is a neighbor of both $v_i$ and $v_j$, and again, using that $i+j$ is minimal we have $i<j<l$. This means that by adjusting the indexes for the remaining vertices in $V_k$ in the natural way ($v_l$ remains $v_l$ if $l<i$, is renamed to $v_{l-1}$ if $i<l<j$ and is renamed to $v_{l-2}$ if $j<l$), the $V_{k-2}$ we got like this satisfies the degree condition $deg(v_i)geq i$ for all index $i$. By induction we are done in this case.

ag.algebraic geometry - Learning about Lie groups

Nobody mentioned "Gilmore: Lie Groups, Physics, and Geometry" yet.

A very down to earth introduction with many examples and clear explanations. Especially targeted at physicists, engineers and chemists.

If you follow the above link you can read some sample chapters.

The cover summarizes the set up of the book quite neatly:

"Describing many of the most important aspects of Lie group theory, this book
presents the subject in a ‘hands on’ way. Rather than concentrating on theorems
and proofs, the book shows the relation of Lie groups with many branches of
mathematics and physics, and illustrates these with concrete computations. Many
examples of Lie groups and Lie algebras are given throughout the text, with applications
of the material to physical sciences and applied mathematics. The relation
between Lie group theory and algorithms for solving ordinary differential equations
is presented and shown to be analogous to the relation between Galois groups
and algorithms for solving polynomial equations. Other chapters are devoted to
differential geometry, relativity, electrodynamics, and the hydrogen atom.
Problems are given at the end of each chapter so readers can monitor their
understanding of the materials. This is a fascinating introduction to Lie groups
for graduate and undergraduate students in physics, mathematics and electrical
engineering, as well as researchers in these fields.

Robert Gilmore is a Professor in the Department of Physics at Drexel University,
Philadelphia. He is a Fellow of the American Physical Society, and a Member
of the Standing Committee for the International Colloquium on Group Theoretical
Methods in Physics. His research areas include group theory, catastrophe theory,
atomic and nuclear physics, singularity theory, and chaos."

at.algebraic topology - Branched coverings over orbifolds with reflector lines

A detailed write up of what you are looking for is also available in Chapter 13 of Thurston's notes. Specifically, formula 13.3.4 on page 331, has the desired formula
$$chi(O) = chi(X_O) -frac{1}{2} sum_{i=1} ^N (1-frac{1}{n_i}) -sum_{j=1}^M(1-frac{1}{m_j}),$$
where $O$ is the orbifold, $X_O$ is it's underlying space, there are N points fixed locally by dihedral groups of orders $2n_1,...,2n_N$ and $M$ points fixed locally by rotations of orders $m_1,...,m_M$.

However, I believe the question addressed (at least in part) by Hurwitz in the proof of Hurwitz's theorem, since the bound the order of an automorphism group H acting on surface of genus g is 168(g − 1) if H includes orientation reversing symmetries and 84(g-1) in the orientable case. The key observation is that the Gauss-Bonnet extends to orbifolds as well.

Computing the Euler characteristic for the quotient of $mathbb{H}^2$ by the (full) (2,3,7) triangle group is a good way to see this in action.

neuroscience - Why are melodies/harmonies perceived as pleasurable by humans?

There are strong connections between the auditory cortex and the limbic system, which includes such structures as the hippocampus and the amygdala.

A recent paper [1] builds on earlier notions of emotional "significance" of music without any lyrics. It adds in lyrics, so giving a perspective of which portions of the brain are reacting to which component of the music.

Additionally, contrasts between sad music with versus without lyrics recruited the parahippocampal gyrus, the amygdala, the claustrum, the putamen, the precentral gyrus, the medial and inferior frontal gyri (including Broca’s area), and the auditory cortex, while the reverse contrast produced no activations. Happy music without lyrics activated structures of the limbic system and the right pars opercularis of the inferior frontal gyrus, whereas auditory regions alone responded to happy music with lyrics.

One of the limitations of this particular study is that the subjects self-selected their own pieces, which may limit the reliability of the results. Of course, defining "happy" or "sad" for every individual is slightly subjective and difficult. They cited an earlier "pioneering" study which standardized the musical selection between subjects. Without consideration of the lyrics:

The first pioneer study using functional magnetic resonance imaging (fMRI) by Khalfa et al. (2005) chose a controlled manipulation of two musical features (tempo and mode) to vary the happy or sad emotional connotations of 34 instrumental pieces of classical music, lasting 10s each. Sad pieces in minor-mode contrasted with happy pieces in major mode produced activations in the left medial frontal gyrus (BA 10) and the adjacent superior frontal gyrus (BA 9). These regions have been associated with emotional experiences, introspection, and self-referential evaluation (Jacobsen et al., 2006; Kornysheva et al., 2010).

As an aside to answer your final thought, in cases like this I think trying to jam everything under an umbrella of one "neurotransmitter system" or another can make things overly simplistic to the point where you lose focus of the diversity of receptors expressed. You can say a system is driven by Dopamine, but D1 and D2 receptors have exactly the opposite effects on the neuron.

[1] Brattico, E., Alluri, V., et al (2011) A functional MRI study of happy and sad emotions in music with and without lyrics. Frontiers in Psychology, 2: 308. doi: 10.3389/fpsyg.2011.00308 (free pdf)

(see also, http://www.sciencedirect.com/science/article/pii/S0028393206003083 and related)

ct.category theory - When does the left-adjoint to a geometric morphism preserve epis?

Suppose I have a functor $f:(C,J)to(D,K)$ between Grothendieck sites. Is there a condition on $f$ such that $f_!$ (the left adjoint to $f^*$) sends "$J$-epimorphisms", to $K$-epimorphisms, where by $J$-epimorphism I mean:

$h:Xto Y$ such that for all $C$, and all $y in Y(C)$, there exists a cover $(g_i:C_ito C)$ in $J$ and $y_i in X(C_i)$ such that for all $i$, $Y(g_i)(y)=h(y_i)$.

EDIT: If X and Y are sheaves, then the notion of "J-epimorphism" coinincides with the categorical epis. As mentioned by David Brown, ANY left adjoint will preserves epis.

In fact, in the situation in which I was interested, I actually have such a (appropriate analogue of a) J-epimorphism between a sheaf and a stack, so, since f_! is a left adjoint, it will preserve this.

ag.algebraic geometry - Infinite projective space

Starting with the affine case, if you try to define infinite dimensional affine space as Spec of k{x₁,x₂,...], then you realise that this is not a vector space of countable dimension, but something much larger. If you want a vector space over k of countable dimension, then this will not be a scheme, but instead will be an ind-scheme. A similar description should hold in the projective case.

Edit: Regarding why I am saying that Spec(k[x₁,x₂,...]) is too big: A (k-)point of Spec(k[x₁,x₂,...]) is an infinite sequence a₁,a₂,... of elements of k. If I wanted a vector space of countable dimension, then I should be asking for sequences a₁,a₂,... of elements of k, only finitely many of which are non-zero. This latter space is the inductive limit of affine n-space as n tends to infinity.

nomenclature - Formal definition of a 'genetic trait reservoir'?

In this context, "trait reservoir" refers to the set of all possible alleles for all the different genes in the organism. The more different alleles the organism has, the more possible genotypes it might have. The Nature paper might have referred specifically to the set of known allelic variants.

nt.number theory - Is there an R=T type result for modular forms with additive reduction?

If you fix a prime $ell$, and consider the Galois action of the decomposition group $D_p$ on the (rational) $ell$-adic Tate module (here "rational" means tensored with $mathbb Q_{ell}$),
then (in a standard way, due to Deligne) you can convert this action into a representation
of the Weil--Deligne group, and so in particular of the Weil group. Restricting to the
inertia group, you get a representation of the inertia group $I_p$, known as the inertial type $tau$. It is independent of $ell$. (The only reason to detour through the Weil--Deligne group is to deal with possibly infinite image of tame inertia; if the elliptic curve has potentially good reduction, then we can skip this step and just take the representation of $I_p$ on the $ell$-adic Tate module, which has finite image and is independent of $ell$.)

[Added: In the above, one should insist that $ell neq p$. If $ell = p$, then one can also arrive at a Weil-Deligne representation, and hence
inertial type, which is the same as the one obtained as above for $ell neq p$, but to do this one must use Fontaine's theory: one forms the $D_{pst}$ of the rational $p$-adic Tate module,
which then can be converted into a Weil--Deligne representation in a standard way,
and hence gives an inertial type.]

Now one can look at the deformation ring $R_{rho}^{[0,1],tau}$ parameterizing lifts
of $rho$ of which at $p$ are of inertial type $tau$ and Hodge--Tate weights $0$ and $1$. (See Kisin's recent
JAMS paper about potentially semi-stable deformation rings.)

[Added: Here $ell = p$,
i.e. we are looking at $p$-adic deformations of $rho$ which are potentially semi-stable
at $p$, and whose inertial type, computed via $D_{pst}$ as in the above added remark,
is equal to $tau$. But note that, by the preceding discussion, these deformations do precisely capture the idea of lifts of $rho$ having the same "reduction type" as
the original elliptic curve $E$.]

Let's suppose that $E$ really does have potentially good reduction. Then Kisin's "moduli of finite flat group schemes" paper shows that any lift parameterized by $R^{[0,1],tau}$ is modular. This shows that $R^{[0,1],tau} = {mathbb T},$ for an appropriately chosen
${mathbb T}$.

One thing to note: unlike in the original Taylor--Wiles setting, from this statement one doesn't get quantitive information about adjoint Selmer groups, and one doesn't get any simple interpretation of what this $R = {mathbb T}$ theorem means on the integral level.
(In other words, Artinian-valued points of $R^{[0,1],tau}$ have no simple interpretation in terms of a ramification condition at $p$; this is related to the fact that the theory
of $D_{pst}$ only applies rationally, i.e. to ${mathbb Q}_p$-representations, not integrally,
i.e. not to representations over $mathbb Z_p$ or over Artin rings.)

algorithms - Effect on connectivity when partitioning a graph

I have a connected graph $G=(V,E)$, $V$ being the vertex set and $E$ being the edge set. I partition the graph into components $C={C_1,dots,C_n}$ such that all $C_i$ are pairwise disjoint.

Take two vertices $s,t in V$ such that $s,t$ are connected by a path. Is there an $O(|V|+|E|)$ algorithm to find out the list of all $C_i in C$ such that if we remove the vertices in $C_i$ from the graph, then $s$ will become disconnected from $t$.

I know there is the $O(|C|(|V|+|E|))$ algorithm to do so by removing vertices in $C_i in C$ from the graph for all $1leq ileq n$ and then checking if $s$ and $t$ are connected.
This can be somewhat improved if we take all edge weights as 1 and compute the shortest path and then consider only components whose vertices are present in the shortest path but this still has worst case complexity $O(|C|(|V|+|E|))$.

dg.differential geometry - virtual bundle with compact support

A virtual bundle with compact support is a triple $E$={$E_1$,$E_2$,$a$},
where $E_1$ and $E_2$ are bundles over $M$ of the same dimension, $M$ is a closed manifold, $a$ is a bundle map from $E_1$ to $E_2$, which is an isomorphism of bundles over $M smallsetminus X$, where $X$ is a compact set in M.

Let $nabla_1$ be a connection on $E_1$, $nabla_2$ a connection on $E_2$. If $X$ is finite number of points in $M$, and Ch($E$) is defined by $text{tr}(exp(-nabla_1)^{2})-text{tr}(exp(-nabla_2)^{2})$, we will find Ch($E$) integrated on $M$ is an integer multiple of a power of $2pi i$.

I want to know, if $X$ is a compact submanifold, then how to get the value of the integration? Maybe we can't get the value but how can we analyze the information about $X$ revealed by the integration?

at.algebraic topology - Maurer-Cartan and representable functors on differential graded commutative algebras

Let $mathfrak{g}$ be a differential graded Lie algebra on a charcteristic zero field, whose underlying chain complex is bounded below and degreewise finite dimensional. Then $mathfrak{g}$ defines a Set-valued functor on differential graded commutative algebras (also, with some finiteness and boundedness assumption) mapping each cgda $Omega$ to the set $MC(Omegaotimes mathfrak{g})$ of Maurer-Cartan elements of the dgla $Omegaotimes mathfrak{g}$ (with the natural dgla structure on the tensor product of a dgla with a cdga). It is well known that this functor is representable: $MC(Omegaotimes mathfrak{g})cong Hom_{dgca}(CE(mathfrak{g}),Omega)$, where the Chevalley-Eilenberg dgca $CE(mathfrak{g})$ is the free graded commutative algebra on the shifted linear dual of $mathfrak{g}$ endowed with the differential induced by the dgla structure on $mathfrak{g}$.

If one looks at the category of commutative graded algebras instead (i.e., one forgets the differential), then $CE(mathfrak{g})$ represents the functor $Omegamapsto (Omegaotimesmathfrak{g})^1$.

My question is: is this latter functor representable also in the category of differential graded commutative algebras? if yes, by which dgca?

gt.geometric topology - CW-structures and Morse functions: a reference request

I'm confused about your 2nd question. Milnor's Morse theory does use a Riemann metric -- he's using the gradient flow. To define the gradient he needs an inner product on the tangent spaces. Without the gradient flow you don't have the cellular attaching maps.

Regarding your 1st question, there's something that's much better than a CW-structure. A Morse function gives a handle decomposition of the manifold. This can be used to talk about the smooth structure. A CW-decomposition is relatively degenerate in comparison. The handle decomposition is described in Milnor's h-cobordism notes.

Taking your 1st question more seriously, you run into technical problems. The gradient flows do not give you a CW-decomposition of the manifold -- for example consider Milnor's Morse Theory example of a torus with height function. The Morse function and its gradient flows gives you a genuine 1-skeleton (figure-8). But the attaching map for the 2-cell (to the figure-8) is not a continuous function if you use the gradient flow -- all points except for two go to the global minimum for the height function. This shows you the kind of problems you encounter if you want to produce a genuine CW-decomposition of the manifold.

So if you're not going to use solely the gradient flows to define the attaching maps for the proposed CW-decomposition, what do you allow? All smooth manifolds admit CW-decompositions so if you allow sufficient tweaking you can of course fix this construction but if you allow "too much" tweaking, the CW-decomposition won't be an invariant of the Morse function.

edit: Here is a way to tweak the process. The gradient flow does give you a genuine 1-skeleton. So take a regular neighbourhood of the 1-skeleton, and perturb the original vector field in this regular neighbourhood to point in towards the 1-skeleton. This makes the 2-cell attaching maps continuous (terminating in a finite amount of time). Then take a regular neighbourhood of the 2-skeleton, and perturb the vector field to point in towards the 2-skeleton. Again, you get flow lines terminating in finite-time so you get genuine 2-cell attaching maps. The problem with this is you're getting a CW-decomposition but it depends on more than the Morse function as you need to choose smooth regular neighbourhoods of the skeleta.

big list - Are there any books that take a 'theorems as problems' approach?

Some answers mention problem books (quite different from standard-format textbooks in that they consist almost entirely of problems and their solutions). Such books have been widely used in Eastern Europe at every level of education (at least when I was getting it). Let me add another one to the list:

MR0447533
Krzyż, Jan G.
Problems in complex variable theory.
Translation of the 1962 Polish original. Modern Analytic and Computational Methods in Science and Mathematics, No. 36. American Elsevier Publishing Co., Inc., New York; PWN---Polish Scientific Publishers, Warsaw, 1971. xvii+283 pp.

In his foreword, the author states: ``Most exercises are just examples illustrating basic concepts and theorems, some are standard theorems contained in most textbooks. However, the author does believe that the reconstruction of certain proofs could be instructive and is possible for an average mathematics student."

Besides standard material, there is a collection of quirky little facts in e.g. non-Euclidean geometry in the disk or logarithmic potential theory (and much more). All stated as problems for the reader to solve. However, many solutions are included.

There were subsequent editions in Polish. I used one as an undergraduate student and still have a copy.

How is the number of mitochondria in a cell regulated?

The concept you refer is recognized as mitochondrial biogenesis and it is regulated by AMPK which senses the cellular energy demand. If you have few mitochondria in the cell, the electron transport chain works suboptminally generating less ATP. When the AMP/ATP ratio is high (low ATP) AMPK is activated, and turns on the catabolic pathways required to produce more ATP, included mitochondrial biogenesis.

linear algebra - Simultaneously orthogonally transform two SPD matrices to tridiagonal form?

Supposing you have two SPD matrices $A,Binmathbb{R}^{ntimes n}$ are there any known results on the existence or non-existence of a unitary matrix $Q$ such that $Q^top A Q=T_A$ and $Q^top B Q=T_B$ are both tridiagonal. If such a transformation exists in general, it is not required for my purposes that it be computable in finitely many steps.

I am aware of non-orthogonal congruence transformations which tridiagonalize two matrices.

Thanks!

---

Edit:

Thanks for the response. I am familiar with the papers of Tisseur and Garvey et. al, but they are using non-orthogonal transformations. In one paper they use alternating 1D Householder reflectors and matrices of the form $L=I+xy^top$ to force portions of the leading columns to be in the same space.

I tried finding a counter-example from the 3x3 case, but it looks like I have plenty of degrees of freedom to play with and higher dimensions become treacherously difficult to manage individual elements.

Maybe this question is equivalent to finding a $Q$ such that for an arbitrary matrix $V$ that $Q^top V$ is bidiagonal, which certainly looks hopeless to me.

intuition - Spectral graph theory: Interpretability of eigenvalues and -vectors

Interesting structural features in graphs are often reflected in eigenvectors (how's that for a generality?) It is often a matter of skill which eigenvector to take. For example, a cyclic grph has multiplicity 2 for the next to largest eigenvalue. Take a 6-cycle (since it is easy). The 2nd eigenvalue is 1 and a generic eigenvector (in cyclic order) is
(1,t,t-1,-1,-t,1-t) SO (looking for relatively few distinct values)
(1,1/2,-1/2,-1,-1/2,1/2) and (1,1,0,-1,-1,0) (and their cyclic shifts) seem nice. Rotating the latter to (0,1,1,0,-1,-1) multiplying by $frac{sqrt{-3}}{2}$ and adding this to (1,1/2,-1/2,-1,-1/2,1/2) does give a nice eigenvector with values on the unit circle, but it is not the eigenvector.

Here is another way to see that $-lambda$ is an eigenvalue of a bipartite graph when $lambda $ is: Take an eigenvector for $lambda$ and replace all the values in one part by their negatives. This shows that (since the largest eigenvalue is unique in that it has an all positive eigenvector) the smallest (in the bipartite case) is unique in that it is positive on one half of the graph and negative on the other, so it reveals the bipartition. One might guess that in a general graph the smallest eigenvalue might have some eigenvectors which partition the vertices into two classes (positive and negative) in a way which minimizes the number of edges connected vertices of the same sign. Several theorems of Miroslav Fiedler are relevant to these considerations.

I'll mention too that I've found it useful to consider the subspace spanned by (eigenvectors for) several eigenvalues. Consider the skeleton of a cube. The sum of the first and second eigenspaces are spanned by the characteristic vectors of the 6 faces. The sum of the first second and third eigenvalues is spanned by the characteristic vectors of the 12 edges. The sum of the first four eigenvalues (i.e. everything!) is spanned by the characteristic vectors of the 8 vertices. This is true (mutatis mutandis) much more generally (Hamming graphs, finite nets, Johnson graphs and other graphs arising from highly regular geometries).

ca.analysis and odes - Are there any nonlinear solutions to $f(x+1) - f(x) = f'(x)$?

Yes, there exist nonlinear solutions.

Multiplying by $e^{x+1}$ and setting $g(x):=e^x f(x)$ transforms the question into finding a solution to $g(x+1)=eg'(x)$ not of the form $e^x(ax+b)$.

Start with any $C^infty$ function on $mathbb{R}$ whose Taylor series centered at $0$ and $1$ are identically $0$, but which is nonzero somewhere inside $(0,1)$. Restrict it to $[0,1]$. Let $g(x)$ on $[0,1]$ be this. Using $g(x+1):=eg'(x)$ for $x in [0,1]$ extends $g(x)$ to a $C^infty$ function $g(x)$ on $[0,2]$, which can then be extended to $[0,3]$, and so on. In the other direction, use $g(x) := int_0^x e^{-1} g(t+1) dt$ to define $g(x)$ for $x in [-1,0]$, and then for $x in [-2,-1]$, and so on. These piece together to give a $C^infty$ function $g(x)$ on all of $mathbb{R}$. The corresponding $f(x)$ satisfies $f(0)=0$ and $f(1)=0$ but is not identically $0$, so it is not linear.

nt.number theory - Can you show rank E(Q) = 1 exactly for infinitely many elliptic curves E over Q without using BSD?

Let $K$ be a number field and let $mathcal O_K$ be the ring of integers. Following this paper of Cornelissen, Pheidas, and Zahidi, a key ingredient needed to show that Hilbert's tenth problem has a negative solution over $mathcal O_K$ is an elliptic curve $E$ defined over $K$ with rank$(E(K))=1$.

Recently Mazur and Rubin have shown that such a curve exists assuming the Shafarevich-Tate conjecture for elliptic curves over number fields. They actually use a weaker, but still inaccessible hypothesis (See conjecture $IIIT_2$).

If you wanted to eliminate the need for this hypothesis you would have to write a proof that simultaneously demonstrated that rank$(E(K))=1$ for infinitely many pairs $(K,E)$ where $E$ is an elliptic curve defined over $K.$ This raises (as opposed to begs) the easier question:

Can you show unconditionally that rank$(E(Bbb Q)) = 1$ for infinitely many elliptic curves $E$ over $Bbb Q$?

It would appear that Byeon, Jeon, and Kim have done so in this paper (probably need an institutional login). Vatsal obtains a weaker result here that still does the job. Unfortunately both of these results invoke the fact that the BSD rank conjecture is true for elliptic curves over $Bbb Q$ with analytic rank 1. Which won't help at present working over number fields.

Can anyone do the above WITHOUT invoking the proven part of the BSD rank conjecture or assuming any conjectures?

pr.probability - Is it a coincidence that the universal parabolic constant shows up in the solution to square point picking?

The "reason" that the two given numbers are equal is "write up the integrals, they turn out to be the same integral".

An answer that might satisfy the "intuitive reason" criterion is that the sides of the unit square are given by axis-aligned straight lines; in polar coordinates $r=frac{1}{cos(t)}$ or $r=frac{1}{sin(t)}$; and $frac{1}{cos(t)^2}=frac{r^2}{r^2cos(t)^2}=frac{sqrt{x^2+y^2}}{x^2}$ is the integrand in the integral defining the arc length of the parabolic segment. Truth be told though, this isn't really much more than saying "the integrals turn out to be the same integral", so I'm not sure how much of an "intuitive" explanation this is.

ho.history overview - Good books on problem solving / math olympiad

From a review for Polya's book on Amazon, the books to be read in sequence:

• Mathematical Problem Solving by Alan Schoenfeld


• Thinking Mathematically by J. Mason et al.


• The Art and Craft of Problem Solving by Paul Zeitz


• Problem Solving Strategies by Arthur Engel


• Mathematical Olympiad Challenges by Titu Andreescu


• Problem Solving Through Problems by Loren Larson

Full text of the review below:

By Abhi:

Good aspects of this book have been said by most of the other
reviewers. The main problem with such books is that for slightly
experienced problem solvers, this book probably does not provide a
whole lot of information as to what needs to be done to get better.
For instance, for a kid who is in 10th grade struggling with math,
this is a very good book. For a kid who is in his 11th grade trying
for math Olympiad or for people looking at Putnam, this book won't
provide much help.

Most people simply say that "practice makes perfect". When it comes to
contest level problems, it is not as simple as that. There are
experienced trainers like Professor Titu Andreescu who spend a lot of
time training kids to get better. There is lot more to it than simply
trying out tough problems.

The most common situation occurs when you encounter extremely tough
questions like the Olympiad ones. Most people simply sit and stare at
the problem and don't go beyond that. Even the kids who are extremely
fast with 10th grade math miserably fail. Why?

The ONE book which explains this is titled "Mathematical Problem
Solving" written by Professor Alan Schoenfeld. It is simply amazing. A
must buy. In case you have ever wondered why, in spite of being
lightning fast in solving textbook exercises in the 10th and 11th
grade, you fail in being able to solve even a single problem from the
IMO, you have to read this book. I am surprised to see Polya's book
getting mentioned so very often bu nobody ever mentions Schoenfeld's
book. It is a must read book for ANY math enthusiast and the math
majors.

After reading this book, you will possibly get a picture as to what is
involved in solving higher level math problems especially the
psychology of it. You need to know that as psychology is one of the
greatest hurdles to over when it comes to solving contest problems.
Then you move on to "Thinking Mathematically" written by J. Mason et
al. It has problems which are only few times too hard but most of the
times, have just enough "toughness" for the author to make the point
ONLY IF THE STUDENT TRIES THEM OUT.

quasi-separated manifolds

Let $M$ be a smooth manifold. Let's call $M$ quasi-seperated if $M$ has the following property: If $B,C subseteq M$ are open balls, then $B cap C subseteq M$ is a finite(!) union of open balls. By an open ball I mean an open submanifold, which is diffeomorphic to some $D^n$.

Is every manifold quasi-separated? If not, are open balls quasi-separated? Is there a simple characterization of quasi-separated manifolds?

Is there a bipartite analog of graph theory?

Some classical theorems involving complete graphs have analogues involving complete bipartite graphs. For example, the complete graph $K_n$ has $n^{n-2}$ spanning trees, while the complete bipartite graph $K_{m,n}$ has $n^{m-1} m^{n-1}$ spanning trees. Finding the largest complete subgraph of a graph is a standard NP-hard problem, and finding the largest (in terms of number of edges) complete bipartite subgraph of a bipartite graph is also an NP-hard problem.

But possibly the area of graph theory that has most benefited from the analogy between general graphs and bipartite graphs is matching theory. Matching theory tends to be easier in the bipartite case, but the bipartite case often gives us clues for the non-bipartite case. For example, finding a maximum matching in a bipartite graph is solvable in polynomial time, but nontrivially so, and this leads us to look for a polytime algorithm for maximum matching in a non-bipartite graph (which does exist, but is more complicated than in the bipartite case). Lovász and Plummer's Matching Theory, now back in print thanks to the American Mathematical Society, gives an excellent account of the interplay between bipartite and non-bipartite matching.

analytic number theory - The Wiener-Ikehara approach to the PNT

This is an elaboration on my comment below John's answer. Its goal is to give a very brief summary of the history underlying the question; I hope that it is more or less correct.

I think that it is fair to say that from the beginning it was understood that non-vanishing on the line $Re(s) = 1$ was the main requirement for proving the prime number theorem.

However, in Hadamard and de la Vallee Poussin's approaches, this non-vanishing was fed into an explicit formula (which gives a Fourier-type expansion of the prime counting function, or some variant), which was in turn obtained from the $zeta$-function by various Mellin transform games. In particular, if I understand correctly, establishing the explicit formula involves pushing the
contour of integration to the left of $s = 1$ (since it involves a sum over the zeroes
of the zeta function), and so doesn't just involve analysis in the
region $Re(s) geq 1$.

As noted in anon's answer, Landau formulated an approach to PNT via a Tauberian theorem involving just analysis on the region $Re(s) geq 1,$ but as well as the crucial condition
of there being no zeroes on the line $Re(s) = 1$, there was a growth condition.

In his paper, Wiener discusses earlier Tauberian approaches, including Landau's, and then
refers to Ikehara's theorem (proved in Ikehara's thesis, I believe, under Wiener's supervision) as being the "true theorem" (I'm fairly confident that I'm remembering his language correctly here), i.e. the one with the correct condition, those conditions being simply that there are no zeroes on the line $Re(s) = 1$.

at.algebraic topology - What is $TC(Sigma^infty Omega X)$?

The TC spectrum, at a prime $p$, of this is the homotopy pullback of a diagram

$S^1 wedge (Sigma^infty_+ Lambda X)_{hS^1} to Sigma^infty_+ Lambda X leftarrow Sigma^infty_+ Lambda X$

after $p$-completion. Here the left-hand map is the $S^1$-transfer from homotopy orbits back to the spectrum and the right-hand map is the difference between the identity and the "$p$'th power" maps on the loop space.

This is in Bökstedt-Hsiang-Madsen's original paper defining topological cyclic homology, in section 5.

ADDED LATER: This doesn't really work on the space level, because they don't have all the structure necessary. They have the $F$ maps, but not the $R$ ones which only come about from stable considerations. Spaces with a group action really only have one notion of "fixed points," namely the honest fixed points of the group action.

However, the associated equivariant spectrum of $Lambda X$ is built out of spaces like

$$Omega^V Sigma^V Lambda X = Map(S^V, S^V wedge Lambda X_+)$$

where $V$ ranges over representations of $S^1$. This has two "fixed-point" objects for any cyclic group $C$: there's the fixed points, which is the space

$$Map^C(S^V, S^V wedge Lambda X_+)$$

of equivariant maps. There is also the collection of maps-on-fixed-points

$$Map((S^V)^C, (S^V wedge Lambda X_+)^C)$$

which is called the "geometric" fixed point object, and it accepts a map from the ordinary fixed points. The fact that $(Lambda X)^C cong Lambda X$ implies that you can interpret this as a map $(Q Lambda X)^C to (Q Lambda X)$ where the latter uses an accelerated circle. These maps give rise to the $R$ maps in the definition of $TC$, and they definitely rely on the fact that you're considering the associated spectra.

st.statistics - Estimate gaussian (mixture) density from a set of weighted samples

The usual EM algorithm can be modified for weighted inputs. Following along the Wikipedia presentation, you would use these formulas instead:

$a_i = frac{sum_{j=1}^N w_j y_{i,j}}{sum_{j=1}^{N}w_j}$

and

$mu_{i} = frac{sum_{j} w_jy_{i,j}x_{j}}{sum_{j} w_jy_{i,j}}$

where $w_j ge 0$ are the weights of the data points.