Let $A = [a_{ij}]_{n\times n}$ be a Hermitian matrix, such that $|a_{ij}| =1$ for $i \neq j$, and $a_{ii} = 0$ for each $i$. I am interested in a tight lower bound of $\|A\|_*:=\sum_{i=1}^n |\lambda_i(A)|$, where $\lambda_i(A)$'s are eigenvalues of $A$.

Note that, by minimizing $\sum_{i=1}^n |\lambda_i|$ over two constraints $\sum_{i=1}^n \lambda_i = 0$ and $\sum_{i=1}^n \lambda_i^2= n(n-1)$, one can obtain $\sqrt{2n(n-1)}$ as a lower bound. But it seems that isn't tight.

On the other hand, if $A := J - I$ (all ones matrix minus identity), then $\sum_i |\lambda_i(A)| = 2(n-1)$.

Is it true that $2(n-1)$ is actually a lower bound (for large enough matrices, say $n \geq 10$) ?

**Added:**

As Alex's answer below, the minimum of trace norm of such matrices may be less than $2(n-1)$, even for arbitrarily large matrices.

@fedja, in a comment below, has been said that the minimum is $(2+o(1))n$ as $n\to\infty$. Could someone give a proof for that?

Let $\{X_t\}_{t\in \mathbb{N}}$ be a strictly stationary and ergodic sequence of real valued random variables and let $X_1$ be supported on $[-1,1]$. Can $(X_1,X_2)$ be supported on the unit disc centered at the origin?

(Under the stronger condition that the sequence is i.i.d., the joint distribution of $(X_1,X_2)$ must be supported on the square $[-1,1]^2$.)

$\bf{Edit:}$ In response to @Nate Eldredge. I will use the following definition of ergodic:

Let $\mu$ be the (shift-invariant) measure induced on $\left(\mathbb{R}^{\mathbb{N}},\mathcal{B}(\mathbb{R}^{\mathbb{N}})\right)$ by the stationary stochastic process $X:=\{X_{t}\}_{t\in\mathbb{N}}$. Let $T:\,\mathbb{R}^{\mathbb{N}}\rightarrow\mathbb{R}^{\mathbb{N}}$ be the left shift operator mapping sequences $\{x_{t}\}_{t\in\mathbb{N}}$ onto $\{x_{t+1}\}_{t\in\mathbb{N}}$.

I will say $\{X_{t}\}_{t\in\mathbb{N}}$ is ergodic if for any measurable $f\in L^{1}(\mathbb{R}^{\mathbb{N}},\mathcal{B}(\mathbb{R}^{\mathbb{N}}),\mu)$, the averages $\frac{1}{T}\sum_{t=1}^{T}f(T^{t-1}X)$ converge pointwise almost everywhere to $\int_{\mathbb{R}^{\mathbb{N}}}f(x)d\mu$.

My 1st question has a straightforward answer but I'd appreciate hints on a proof. My 2nd question is open from my point of view.

** Q1**. Is it the case that the maximum

Let $C$ be a smooth curve in $\mathbb{R}^3$, whose maximum curvature at any point $x \in C$ is $\le 1$. Now consider a tubular neighborhood of $C$— (used also in Light rays bouncing in twisted tubes)— width of $r<1$.

** Q2**. Let the curvature of the smooth $C \in \mathbb{R}^3$ be bound by $\le 1$.
What is (a description of) the maximum volume convex shape that could move
(via rigid motions) through

A smooth curve $C$ with curvature everywhere $\le 1$. Tube of radius $r < 1$.

I presume the optimal shape is convex. I suspect this question has been considered previously...?

Related: Sofa in a snaky 3D corridor.

Let $G$ be a torsion-free group. For an element $g \in G \setminus \{1_G\}$ we define the radical of $g$ in $G$ as $$ Rad_G(g) = \left\{r \in G \mid r^a = g^b \mbox{ for some } a,b \in \mathbb{Z}\setminus\{0\}\right\} \cup \{1_G\}. $$

For which torsion-free groups can we show that $Rad_G(g)$ is an infinite cyclic subgroup of $G$ for every nontrivial element? So far I have been able to establish it for the following classes:

- residually finitely generated torsion-free nilpotent groups,
- torsion-free hyperbolic groups,
- relatively hyperbolic groups (in the sense of Bowdich), where the associated subgroups already have the property (this includes for example toral relatively-hyperbolic groups).

Are there any easy examples I am missing?

$\DeclareMathOperator\Tr{Tr}$Say that we have an algebra $R$ over $\mathbb{C}$. If, for two finitely generated $R$-modules $M, N$ we know that $\Tr_M(r)=\Tr_N(r)$ for all $r\in R$, where we consider $r$ as the linear endomorphism of the corresponding module, then we know that $M\cong N$ by the Brauer–Nesbitt theorem.

My question is, assuming that we have specific values for traces (some homomorphism $g: R \rightarrow \mathbb{C}$ with $g(rr')=g(r'r)$ for every $r, r'\in R$), do we know that there exists some module $M$ with $Tr_M(r)=g(r)$ for every $r$? If not, can we somehow efficiently describe the functions that can appear as traces?

In their paper, Kronheimer and Mrowka constructed an instanton homology $J^{\#}$ for webs and foams and conjectured that for planar webs, $\dim J^{\#}=\#\text{ of Tait colorings}$. According to my limited understanding, $J^{\#}$ is a sort of a TQFT (with $\mathbb{F}_2$ coefficient), or a functor from the category of webs and foams to the category of $\mathbb{F}_2$ vector spaces and morphisms.

There is a familiar TQFT in one lower dimension which is clearly related to the number of Tait colorings : namely, the (Chern-Simons) quantum invariant for the second symmetric representation of $SU(2)$. In the classical limit $q=1$, it is exactly the number of Tait colorings.

I know this is probably just a far-fetched speculation, but I am curious if these two TQFTs can be somehow related. More precisely, **is it possible to categorify the latter such that it becomes $J^{\#}$ when reduced to the $\mathbb{F}_2$ coefficient?**

Let $X$ be a smooth and projective variety of dimension $d>1$. Let $X^{[2]}$ denote the Hilbert scheme of length two subschemes of $X$. Let $X^{(2)}:=X\times X/\mathbb{Z}_2$, where $\mathbb{Z}_2$ acts by $(x,y)\mapsto (y,x)$. Then there is a birational map $X^{[2]}\to X^{(2)}$. Let $E$ denote the exceptional divisor if this map. Or $E$ can be described as the divisor whose locus is the set of non-reduced subschemes. Can someone point a nice reference where it is explained that there is a line bundle, whose square is the line bundle corresponding to $E$.

Alternatively, the question can be posed as follows. Let $Y$ denote the blow up of the diagonal of $X\times X$. Then the action of $\mathbb{Z}_2$ extends to $Y$, (I think this action is trivial when restricted to the exceptional divisor). The quotient is $\pi:Y\to X^{[2]}$. How do I see that there is a divisor $F$ on $X^{[2]}$ such that $2F=\pi_*(E)$?

I am looking for a reference or explanation of this fact.

Let $E$ be a quadratic extension of a local nonarchimedean field $F$ of characteristic zero (and odd residual characteristic). Let $\sigma$ be a generator of the Galois group $G = Gal(E/F)$. I'm looking for an elementary proof that there are infinitely many distinct (unitary) characters $\chi$ of $E^\times$ such that ${}^\sigma \chi = \chi \circ \sigma \neq \chi$.

Here's a sketch: the character $\chi$ is Galois invariant, i.e., ${}^\sigma \chi = \chi$ if and only if $\chi$ is trivial on the kernel of the norm map $N_{E/F}: E^\times \rightarrow F^\times$. Let $K = \ker N_{E/F}$. The group $K$ is a closed subgroup of the compact group $U_E$ of units in $E^\times$. We can extend any non-trivial character $\tilde \chi$ of $K$ to $E^\times$ to obtain a nontrivial character $\chi$ of $E^\times$ such that ${}^\sigma \chi \neq \chi$.

The part of the argument that is missing is to show that either:

(a) the character group $\widehat K = Hom(K,S^1)$ of $K$ is infinite,

or, if (a) is false (?),

(b) if $\widehat K$ is finite, then we need to show that there are infinitely many distinct extensions of at least one $\tilde \chi \in \widehat K$ to $E^\times$

I expect that (a) is true. Any suggestions to address this fact (or a reference) would be greatly appreciated.

In recent question we have discussed a coeffcients $A_{m,j}$, such that for every integer $n\geq0$ we have identity \begin{equation} n^{2m+1}=\sum\limits_{1\leq k \leq n}D_{m}(n,k) \end{equation} where $D_m(n,k)$ is defined by \begin{equation*} D_{m}(n,k):=A_{m,m}k^m(n-k)^m+A_{m,m-1}k^{m-1}(n-k)^{m-1}+\cdots+A_{m,0} \end{equation*} Coefficients $A_{m,j}$ in above definition are terms of OEIS sequences A302971 and A304042 for numerators and denominators of $A_{m,j}$, respectively. Consider a few examples for some positive integers $m,n$. Let be $m=3, \ n=4$, then \begin{eqnarray*}\label{gen_22} 4^{2\cdot3+1} &=&1-14\cdot3^1+0\cdot3^2+140\cdot3^3 \\ &+&1-14\cdot4^1+0\cdot4^2+140\cdot4^3\\ &+&1-14\cdot3^1+0\cdot3^2+140\cdot3^3\\ &+&1-14\cdot0^1+0\cdot0^2+140\cdot0^3\\ &=&3739+8905+3739+1=16384 \end{eqnarray*} Where coefficients $\{A_{3,j}\}_{j=0}^{3}=\{1,-14,0,140\}$ are terms of third row of A302971 and $\{3,4,3,0\}$ are terms of forth row of triangle A094053. Similarly, let show example for $m=4, \ n=5$, we get \begin{eqnarray*}\label{gen_24} 5^{2\cdot4+1} &=&1-120\cdot4^1+0\cdot4^2+0\cdot4^3+630\cdot4^4 \\ &+&1-120\cdot6^1+0\cdot6^2+0\cdot6^3+630\cdot6^4\\ &+&1-120\cdot6^1+0\cdot6^2+0\cdot6^3+630\cdot6^4\\ &+&1-120\cdot4^1+0\cdot4^2+0\cdot4^3+630\cdot4^4\\ &+&1-120\cdot0^1+0\cdot0^2+0\cdot0^3+630\cdot0^4\\ &=&160801+815761+815761+160801+1=1953125 \end{eqnarray*} As it in previous example, the terms are of sequences A302971 and A094053 with respect to $m,n$. We can observe that the number of lines in each example is $n$ and the number of terms in each line is $m+1$, therefore, we can assume that result of each example could be reached by operations on certain $n,m-1$ dimension matrices. Hence, the question stated

**Question**: Is it possible to implement above method in terms of certain matrices? And, if yes, how?

Mathematically the definitions are as follows : if $H_n$ is a $n-$dimensional complex Hilbert space then its two different corresponding ``Fock space"(s) are often denoted as $F_{1}$ and $F_{-1}$ defined as, $F_1 = \oplus_{k=0}^{\infty} Sym^k(H_n)$ and $F_{-1}= \oplus_{k=0}^{\infty} \Lambda^k(H_n)$.

Physically for a Quantum Field Theory one "defines" its so-called Hilbert space as the dual of an implicit vector space over $\mathbb{C}$ whose basis is in bijective correspondence to the set of all possible values for all the classical fields that occur in the underlying Lagrangian.

Now my question is two fold,

Does this physical notion of a "Hilbert space of a QFT" correspond to the $H_n$ or some ``total Fock space" that can be defined from the first mathematical definition as, $\otimes_{i \in Fields} F^i_{p_i}$ where $p_i=$1 if the $i^{th}$ field is Bosonic or $-1$ if it is Fermionic? (..I guess this tensoring is needed because the QFT can have both Fermionic as well as Bosonic fields..)

If we agree as above that the states of a QFT live in such a "total Fock space" and not in the Hilbert space defined in the second paragraph then shouldn't the "Quantum Field" be mapping into a space of Hermitian operators on this total Fock space and not just the Hilbert space?

We are studying the behavior of families of curves inside stable families of surfaces. The non-existance of the following configurations of curves in a non-normal surface would be sufficient to prove our result.

Let $X$ be a non-normal surface over $\mathbb{C}$ with non-normal locus (i.e. double locus) $X_{dl}$.

Let $D \subset X$ be a nodal curve, with node $p$ so that $p \subset X_{dl}$, the pair $(X,D)$ has semi-log canonical singularities, and $K_X + D$ is ample.

Let $C = C_g \cup C_0$ be a genus $g(C) = g$ irreducible curve, which is the union of a genus $g$ curve $C_g$ and a rational curve $C_0$. Further suppose that $C_g \cap C_0 = p$ (the node of $D$), and that $C_0 \subseteq X_{dl}$.

Finally, suppose that $C_0 \cap D = p$ and the normalization of $X$ is irreducible.

Is such a configuration possible? For instance, is $C_0 \cap D$ forced to be more than one point?

I am looking at the first proof of the existence of a fundamental solution for Linear partial differential equations with constant coefficient (The 3.1.1, Linear Partial Differential Operators, Hormander, Springer Verlag 1964), using Hahn-Banach. In order to construct a fundamental solution such that $$ P(D) E = \delta $$ Hormander uses a lemma giving for $u$ a smooth function with compact support : $$| u(0) | \leq C || P(D)u || .$$ He constructs a function $E \ast \delta_0$ which is defined as $P(D) u \mapsto u(0)$ on the vector space of functions $\{g | g = P(D)u \text{ for some } u \}$ and then extended to $E \ast \delta_0 : \mathcal{D}(\mathbb{R}^n) \to \mathbb{R} $ by Hahn-Banach. My naive question is how is $E \ast \delta_0$ well defined in the first place, as when considering a function $f = Du$, $u$ and thus $u(0)$ may vary ? I agree that once $E$ is fixed then $u$ and $u(0)$ are unique, but I don't see how this can be at the beginning of the proof. Is any of you familiar with this proof ? Am I missing something obvious ?

Thanks.

The finite field of order $p^n$ is isomorphic to $(\mathbb Z/p \mathbb Z)[X]/(P)$, where $P$ is an irreducible polynomial in $(\mathbb Z/p \mathbb Z)[X]$ of degree $n$. This describes every finite field up to isomorphism.

My question is, what are alternate ways of describing these finite fields?

For example, Conway gave an alternate way of describing the fields $GF(2^{2^n})$ (see this example answer).

Say $L^2(M_1,\mathbb{R})$ is the space of square-integrable functions on $M_1$ which is a compact manifold with a measure defined on it. Let $L^2(\mathbb{R}^n,\mathbb{R})$ be the space of all square-integrable functions mapping $\mathbb{R}^n \rightarrow \mathbb{R}$.

Now I consider the space of all functions $F$ mapping, $M_1 \rightarrow \mathbb{R}$ of the form, $\{ f(g_1,..,g_n) \vert f \in L^2(\mathbb{R}^n,\mathbb{R}) \text{ and } g_i \in L^2(M_1,\mathbb{R})\}$

Now given a (countable) basis for $L^2(M_1,\mathbb{R})$ and $L^2(\mathbb{R}^n,\mathbb{R})$ can one write down a countable basis for $F$?

Are there general theorems known about when one can write down a basis for the composed space in terms of a given set of bases for the spaces being composed?

Task question:

Please solve and visualize movement using winbgi.h graphic library (available from faculty/AeroDiv/Courses website).

• dx/dt = -y - z • dy/dt = x + ay • dz/dt = b + z(x - c) • Usual parameters: a = b = 0.2, c = 5.7

Visualize z=z(t,x,y). Below is the code for the above problem :

include include include include include include"winbgi2.h" define n 3 define dist 0.1 define max 50void rungekutta(double x, double y[], double step);
double fun(double x, double y[], int i);
void main()
{
graphics(50,30);
double t, z[n];
int j;
FILE *f;
fopen_s(&f, "padela.txt", "w+");
t = 0.0;
z[0] = 0;
z[1] = 0;
z[2] = 0;
fprintf(f,"time\t\tZ-axis\n");
fprintf(f, "%lf\t\t%lf\n",t,z[2]);
for (j = 1; j*dist <= max; j++)
{
t = j*dist;
rungekutta(t,z,dist);
fprintf(f, "%lf\t\t%lf\n",t,z[2]);
circle(t+200,z[2]+200,7);
}

fclose(f); getch(); }

void rungekutta(double x, double z[], double step)
{
double h = step / 2.0,
t1[n], t2[n], t3[n],
k1[n], k2[n], k3[n], k4[n];
int i;
for (i = 0; ifun(x, z, i));
}
for (i = 0; i(k2[i] = step*fun(x + h, t1, i));
}
for (i = 0; ifun(x + h, t2, i));
}
for (i = 0; i
*

*double fun(double x, double z[], int i)
{
double a=0.2,b=0.2,c=5.7;
if (i == 0)
{
return -z[1]-z[2];
}
if (i == 1)
{
return z[0]+(az[1]);
}
if (i == 2)
{
return b+(z[2]*(z[0]-c));
}
}
I need the continuation code for the program which states below question:
1) Find the peak value of Z from Z-t graph at 43 second. (Since the h is changing, so find the peak value at 43 second) consider the peak value as ZR.
2) Calculate value of Z (h) where h= 1/100000. The values of h is multiplied by 10 every time till h=1.
3) Calculate the error. Error = (Z (h) - ZR) / ZR
4) If the value of error in the range 2*10-3 < …….<4*10-3. Then print the value of error and also h (step).

Let $E$ be an infinite-dimensional complex Hilbert space.

For $T = (T_1,\cdots,T_n)\in\mathcal{L}(E)^n$, the algebraic spectral radius of $T$ was given by $$ r_a(T_1,\cdots,T_n)=\lim_{k\to+\infty}\left\|\sum_{f\in F(k,n)} T_f^* T_f\right\|^{\frac{1}{2k}} , $$ where $F(k,n):=\{f:\,\{1,\cdots,k\}\longrightarrow \{1,\cdots,n\}\}$ and $T_f:=T_{f(1)}\cdots T_{f(k)}$, for $f\in F(k,n)$.

Gelu Popescu has a paper (Memoirs of the AMS, arXiv). In page 8 of this memoirs I see the following paragraph:

Are the two definitions equivalents?

A question I was asked recently lead me to the following question. For every closed first order formula $\theta$ in the group signature consider the set $N_\theta$ of natural numbers $n$ such that the symmetric group $S_n$ satisfies $\theta$.

**Question.** Can $N_\theta$ always be defined by a first order formula in the signature of natural numbers?

**Update.** The original question has been answered below by Noah Schweber, but it occurred to me that I am mostly interested in the converse translation. So here is

**Converse question.** Given a first order formula that defines a set $M$ of natural numbers, is there always a first order formula in the group signature defining the set of symmetric groups $\{S_n\mid n\in M\}$?

Suppose we have an M$\times$N complex matrix $H$ and its singular value decomposition $H=U\Lambda V^*$ and an N$\times$N covariance matrix $R_s$ with its eigendecomposition $R_s = U_s\Lambda_sU_s^*$. We also have the eigendecomposition of $HR_sH^*$ as $U_A\Lambda_AU_A^*$. In my research problem setting, I know $U_A$, $H$ but not $R_s$ and $\Lambda_A$. I want to find $U_s$ using $H$ and $U_A$. I was wondering if there exists any connection between them.

I am trying to prove Proposition 2.1.1 of Gauduchon's note on Kähler extremal metrics (page 67). In order to show that, for compact Kähler manifolds, the complex Lie algebra of real holomorphic vector fields $\mathfrak{h}$ is finite-dimensional, he argues the following.

Firstly $\mathfrak{h}$ is identified with the space of holomorphic vector fields of $T^{1,0}$, which is equipped with an obvious $\overline{\partial}$ operator. Then he claims that holomorphic vector fields belong to $Ker(\overline{\partial})$ and $Ker(\overline{\partial}^*)$, being $\overline{\partial}^*$ the adjoint of $\overline{\partial}$, with respect to any compatible metric. Finally, since the kernel of the elliptic operator $\overline{\partial}^*$ + $\overline{\partial}$ is clearly finite dimensional, the result follows.

I am not able to prove that an holomorphic vector field $X$ belongs to $Ker(\overline{\partial}^*)$. When I identify $X$ with a (0,1)-form $\omega$, using the Kähler metric, it is not hard to prove that $\omega$ is $\overline{\partial}$-closed. However, expressing $\omega$ in local coordinates, I am not able to derive also the $\overline{\partial}^*$-closedness.

Do you have any suggestion?

If $Y\subset X^*$ is a closed subspace (where $X$ is a separable Banach space), the preannihilator of $Y$ in $X$ is $Y_{\perp}:=\{x\in X : y^*(x)=0, \forall y^*\in Y \}$. If $Y$ is a proper subspace of $X^*$, and $X$ is reflexive then it can be proved that $Y_{\perp}\neq\{0\}$. On the other hand if $X=l_1$ and $Y=c_0\subset l_{\infty}$, then $Y_{\perp}=\{0\}$.

- 1) If $X^*$ is separable, is it still possible that $Y_{\perp}=\{0\}$?
- 2) Can $c_0$ be a subspace of a separable dual? This question is probably unrelated, and most likely well known, but I realized I do not know an answer to it.