Here’s a new Graduate Research Fellowship Program (GRFP) supported by NSF. I think Math PhD candidates may have interest in it.

**Application Deadline(s)** (received by 8 p.m. Eastern Standard Time):

November 05, 2013

Mathematical Sciences; Chemistry; Physics and Astronomy

#### General Information

**Program Title:**

NSF Graduate Research Fellowship Program (GRFP)

**Synopsis of Program:**

The purpose of the NSF Graduate Research Fellowship Program (GRFP) is to help ensure the vitality and diversity of the scientific and engineering workforce of the United States. The program recognizes and supports outstanding graduate students who are pursuing research-based master’s and doctoral degrees in fields within NSF’s mission. The GRFP provides three years of support for the graduate education of individuals who have demonstrated their potential for significant achievements in science and engineering research.

**Cognizant Program Officer(s):**

*Please note that the following information is current at the time of publishing. See program website for any updates to the points of contact.*

- Applications, contact: GRF Operations Center, telephone: (866) 673-4737, email:info@nsfgrfp.org

- Gisele Muller – Parker, telephone: (866) 673-4737, email: info@nsfgrfp.org

- Doris Carver, telephone: (866) 673-4737, email: info@nsfgrfp.org

- Pushpathala Murthy, telephone: (866) 673-4737, email: info@nsfgrfp.org

For details, please click this linkage http://www.nsf.gov/pubs/2013/nsf13584/nsf13584.htm.

here,i list the 18 proplems i proposed in 2012,which are the conclusion of 10 mathematical papers of mine(online papers). in these 18 problems, i sketch an outline for some technique difficulties we meet in topology,analysis,pdes and even algorithms.i hope visitors will read them for me and give me advise for whether these research plans are feasible.

PDEs(navier-stokes equation)

problem 1:

how to use the identity representation and the volumn to study the relation between the viscosity and the eigenvalue?

here,we study the symmetric solution of the navier-stokes equation.firstly we get that under what condition, when the viscosity convergence to $ infty $ the eigenvalue vonvergence to $ infty $ .this condition is related with the function that $ u^2 + 3/2 u -(4 gamma -3)/4 =0 $ in which we use the isentropic property of the navier-stokes equation，and we also consider the condition of the delta $ Delta $ for the eigenvalue.

then we get the inequality about the identity representation and the volumn by applying the property of the symmetric solution , such that:

$ rVert {1Q^2 Q} rVert ge 1 $.

our problem is that how to use the inequality $ rVert {1Q^2 Q} rVert ge 1 $ to study the relation between the viscosity and the eigenvalue.

you can go to this url: vixra.org/abs/1302.0024 page12-24

vixra.org/abs/1304.0057 problem 8,9 and page 11 conclusion 4

problem 2:

the velocity ratio for the symmetry of the L$(3,infty)$ case ?

in the navier-stokes equation, the Ls case such that s is included in (3,$infty$) is studied by ladyzhenskaya.in our research, we discover a new symmetry in the Ls case that when the parameter C is equal to $[(1/2)^3/^2]/10$,the L3 space holds; otherwise when the parameter C is equal to $-[(1/2)^3/^2]/10$, the L infinite space holds;

our problem is that how to apply this symmetry. a practible way we propose is that we can use the velocity ratio to construct the Ls type space. and our method is based on an inequality that the integral of the velocity $int v^2$ is less that $v^2$, eventually we get this inequality by using the hermit polynomial.

for more detail,please go to this url: vixra.org/abs/1302.0024 page2-12

vixra.org/abs/1304.0057 problem 6,7 and page 10-11 conclusion 3

problem 3:

how to use the boundary viscosity to study the sound speed integral function $ k(y) $ ?

step 1 :we use bessel function to get the boundary condition of the viscosity.

step 2 :we use a lemma in heaviside function to get the condition of when the entropy flux kernel convergence to zero, then the integral function of the sound speed k(y) convergence to zero.

step 3 :we construct euler substitution to get different inequality for the bound of the function f(x) by introduce the ellipsoid coordinate.

finally,from step 1 we can get the boundary viscosity,from step 2 we can use the entropy flux kernel to omit the infinite small in euler substitution.then we can construct different euler form function to achieve boundary control towards the integral function of the sound speed $ k(y)=int c(y)/y $.

for more detail, please go to vixra.org/abs/1302.0025 page16-29

vixra.org/abs/1304.0057 problem 3,4,5 and page9-10 conclusion 2

problem 4:

a criterion for the initial viscosity and the limit viscosity?

here we use a well known intergral inequality in navier-stokes equation to get a criterion about the limit viscosity and the initial velocity and initial viscosity.our procedure is:

1.get a estimate for the velocity which is based on the parameter $2k0(1/4π)^2$ . this estimate relate the laplacian operator (for the velocity) and the M (the integral kernel of the viscosity)

2.to estimate the external force in which we get an unique bound for the force f,and the bound is $c^2$,that means we can use the bound to omit the surplus terms in the inequality

which we mentioned at the beginning.

3.use the parameter $2k0(1/4π)^4$ to study the relation among the limit viscosity and the initial velocity and initial viscosity.

for more detail,you can refer to: vixra.org/abs/1304.0057 (page 8-9 conclusion 1)

problem 4.1:

a new method to estimate the external force by applying a special discrete function?

firstly,we construct a map from [0,π] into [0,1],to introduce the function x(t),by immediate calculation we can get an estimate with constant C0 and C1.then we undetermine the coefficient to seach the value of C0,C1 so that we can estimate the operator function $d(x)$,then to estimate the external force.

in our study of the operator function $d(x)$,we discover that under some assumption we can get an unique bound for the force f ,which is formed by the constant $c^2$.

for more detail you can go to this url: vixra.org/abs/1302.0025 page 10-16

vixra.org/abs/1304.0057 (problem 2)

problem 4.2:

is this new velocity estimate for weak solution useful?

here,we raise a new method to estimate the velocity for weak solution.our procedure is:

1.study the laplacian operator,to get the condition that relate the time t and M (the integral kernel of the viscosity)，and also prepare to transform the two dimension coordinate to three dimension coordinate.

2.transform the equation to poisson form and also shift the coordinate,to set the initial radius equal to zero and get the last estimate.in which we use a lemma in integral equation so that we can apply the spherical coordinate to estimate the value of $cos a$ in the poisson solution.

our last estimate for the velocity is based on a parameter such that $2k0(1/4π)^4$ ,which is a valuable result can be used in fluid mechanics(k0 is a max function about the initial velocity and the initial viscosity).

another technique problem we mentioned is that we can use the well known lemma in integral equation(we use in step 2),to get a more practical condition the velocity estimate need, which is a relation among the time t and the initial velocity and the initial viscosity.

for more detail,please goto the url: vixra.org/abs/1302.0025 page 3-10

vixra.org/abs/1304.0057 (problem 1)

topology(convex cone)

problem 5:

orthogonal condition and the angle 45 degrees of convex cone?

we discover that the orthogonal condition is related with the angle 45 degrees for convex cone. and we also cite three examples about the application of the angle 45 degrees.the three examples are:

1.real symmetric orthogonal matrix

2.bounded norms in vector spaces related with the function xsin(1/x),the function is used to study the angle 45 degrees of convex cones and conversely we can use the property of convex cone to study the positive and negative to the continuous function

3.by studying the positive root of the quadratic function we can verify the relation between the orthogonal condition and the angle 45 degrees,conversely we can use this example to study the quadratic function associated with convex cone.

for more detail,you can refer to the url: vixra.org/abs/1302.0050 page1-13

problem 6:

how to calcular the included angle by studying the circle norms?

in convex cone we always construct the circle norms ,for example:

$x1^2+x2^2>(1/9)$,$(x1-(1/2))^2+(x2-(1/2))^2>(1/9)$,$x=a(1,x2,x3,…)+b(-1,x2,x3,…)$

our method is to analysis the closed ball about the nontrivial convex cone in a normed space and get a contradiction first, in which we get a relationship between the angle and the parameter 1/k,1/n. then we use tools in plane geometry and analytic geometry to study the circle norms and get the relationship between the included angle and the position angle.our main problem is that how to combinate step 1 and step 2,so that we can determine whether the closed balls are diviation or tangent.

for detail you can refer to the url: vixra.org/abs/1302.0050 page13-19

PDEs(solitary waves)

problem 7:

can we approximate to the singular point in analytic function?

if we can use the set $W=(w-(m+x)^1/^2)le epsilon $

to study the analytic function,can we construct an equivalent set ,for example the set $V=(n+1)^1/^2-n^1/^2le epsilon $ which is equivalent to the set W, to achieve the topological equivalence which can omit the infinitely small and lead that we can get the the upper bound of the integral equation.

to the cos case ,the upper bound is 1+i/2;

to the sin case ,the upper bound is i+1/2i.

if we can approximate to the singular point, what is this symmetry imply? how to apply this symmetry to the problems we meet in the klein-gordon equation?

for detail you can refer to the url: vixra.org/abs/1301.0169 page13-17

problem 8:

the convergence of the series in the klein-gordon equation?

here, i list a new method to study the convergence of the series in the klein-gordon equation.

firstly, transform the equation to the form $Cj/(kj+ka)e^-(kj+ka)x$ by analyze the matrix.

then,construct integral kernel to make it suit the form we get in the first step and also suit the standard form of the integral kernel equation.

finally,by comparing the value of $C=Ym+1$ we can determine the convergence of this type equation in which we use the chebshev inequality to get the value C.

is this method feasible?

for detail you can refer to the url: vixra.org/abs/1301.0169 page17-21

analysis(p-adic)

problem 9:

how to construct equal orthogonal bases?

can we get equal orthogonal bases by searching the optimal approximation to the multi-orthogonal bases? and our searching methods is basing on a generalization of the non-archimedean norm spaces.

for more details, you can refer to the url : vixra.org/abs/1302.0007 (page 2-13),the paper named “non-archimedean analysis-the application of symmetric methods”

our step is as follow:

1.generalize the symmetric norms to n dimension.

2.estimate the max and min value of the parameter a.

3.by add or substract terms at the right hand side of the inequality(which we get by the estimate in step 2) to achieve the optimal approximate towards the bases, which relate step 1 and step 2,then we can get equal orthogonal bases under the condition of limit and the estimate is optimal.

problem 10:

a new relationship between cartesian and hilbertian?

a norm space，if its every finite dimension linear subspace has orthogonal base，we call it Cartesian. If its every one dimension subspace has orthogonal complement，that is Hilbertian. If a norm space is Hilbertian，then it is Cartesian. But, if a space is cartesian,is it Hilbertian? In the complete case, this composition is true. But, in the dense case, that is uncertain. here,i list a new relationship between cartesian and hilbertian .i wrote this relationship in the paper “some problems on orthogonal cartesian spaces “and another paper named “nonarchimedean analysis-the application of symmetric methods” on vixra ,the url is: vixra.org/abs/1302.0007 vixra.org/abs/1301.0181, and our procedure is : step 1.find the growth mode of the theradius of the closed ball and use the the modulus n remainder k group to represent the theadius(page 10 to 13 in “some problems on orthogonal cartesian spaces” ) .step 2.use the helly theorem to get the relation estimate (page 9 formular (11) in “some problems on orthogonal cartesian spaces” ) step 3.use an inequality to relate the group in step 1 and the estimate in step 2. then we should consider the limit form of the estimate we get in step 2 by applying the inequality we use in step 3,which lead that the c and the k in the estimate(step 2) are equal when they converge to infinite in the meaningful of limit , so we can get the last estimate (refer to the example five page 20 to 23 in “nonarchimedean analysis-the application of symmetric methods”) which relate cartesian and hilbertian.

problem 11:

how to separate the variables in the max function in p-adic?

i discover a new method to separate the variables in the max function we meet in p-adic. for example,

vixra.org/abs/1302.0007 page23-27

firstly,we use a parameter t to represent the form of (a1+v0)/a2

then,we apply the symmetry of the norms f0,f1 to achieve variable separation towards the max function

algorithms:

problem 12:

is there a way to estimate the meanvalue basing on orthogonal bases(p-adic)?

The main procedure is that: firstly we cut off finite points from the sequence. Then by using these finite points, we can define intervals, the intervals can be fraction(denominator are added by integer); They can also be continued fraction. Then，we apply the scaled methods, firstly we should determine the value of two ratios r and r’（pay attention that : when we calcular the value of r and r’ ,we can omit the repeated items）Then，by using this ratios we can determine the size of meanvalue.

two crucial points we should mention is that :in this algorithm we should search another dn for comparison and we use the orthogonal base to define intervals at first.also note that the one to one correspondence between the definition and the inequality which we use to determine the meanvalue.

the paper is online: vixra.org/abs/1301.0181 named “some problems on orthogonal cartesian spaces” page 13-28

problem 13:

a phenomenon in simplex linear programs?

firstly,we define the ideal status of simplex program such that we can calcular the optimal value for the simplex in linear programs by setting the main element 0 and setting the main elements anfirstly,we define the ideal status of simplex program such that the primary base are formed by 0 and 1.

then we make assumption d auxiliary elments equal.

for detail you can refer to the url: vixra.org/abs/1302.0057 page3-7

problem 14:

can we view the hanoi tower algorithm as the following way?

the well known hanoi tower algorithm is as follow:

public static void hanoi(int n,int a,int b,int c)

(

if(n>0)

(

hanoi(n-1,a,c,b);

move(a,b);

hanoi(n-1,c,b,a);

)

)

my problem is : can we handle this algorithm in this method?

we can regard three places as one place which means that three places’ case is the basic case,then we can solve the problem of nine places where three places are regarded as one place, and so on, we can solve the case of 27 places then to the case of $3^n$ places. if we realized that when we move three places, it means that we move 7 times which time we move one place,in this way we can solve the problem with remainder.

i list this problem in a paper of mine named “lecture notes on recursive algorithm “which is on line on vixra.org,the url is: vixra.org/abs/1303.0106 (part 3.1)

ohter fields(real analysis,fixed point theory,algebraic geometry,java)

problem 15:

a procedure in the finite subcovering theorem?

we discover a new relation between the finite subcovering theorem and a well-known theorem such that : there did not exist any lebesgue measurable set E included in [0,1], for all (a,b) included in [0,1] , and denote that the measure of the intersection between E and (a,b) is equal to (b-a)/2.

the relation imply that to every finite subcoverings . if it is lebesgue measurable set , then to the complement sets of the coverings (the coverings except the neighbourhood

of x ) ,the procedure mentioned can be continued.

and the sequence is 1/3,1/2,5/6 ,eventually the sequence will repeat again and again.

for detail you can refer to the url: vixra.org/abs/1302.0057 page7-10

problem 16:

can we use the weak fixed point property to handle the schauder conjecture?

the schauder conjecture that every compact convex subset of a metric space has a fixed point was established by cauty in 2001. my problem is that: can we use the weak fixed point property to handle the schauder conjecture? i list the procedure in the paper :“weak fixed point property and schauder conjecture ”on vixra and the url is : vixra.org/abs/1306.0199. i hope visitors can read it for me and answer me whether this procedure is feasible?

here i list the general ideal and the procedure:

1.use the error estimate to study the separable property.

2.to get an inequality relate the p in weak fixed point property with the bound for the projection operator.

3.to get an inequality exists in some special dimension about the schauder conjecture.

4.use the inequality we get in step 3 to relate the separable property (step 1)with the parameter k in weak fixed point property by characterize the compact group.

5. to different dimension we get different bound of the k in the weak fixed point property . maybe ,i guess that the bound of the parameter k in weak fixed point property support that we have a fixed point to the metric space with compact convex subsets.

problem 17:

can we apply the property of the line at infinity to the residues at infinity?

as is well known, the line at infinity has marvellous property which are widely used in conic.my problem is that can we apply this property to residues?

for more details,you can refer to the paper “interesting mathematics ”(example one)and the url is ：vixra.org/abs/1302.0057

to the residues, an important property is that :when we multiple the equation with i. It means that we can turn the whole closed neighbourhood over.another point can not be omit is that we can imagine the closed neighbourhood of the residues at infinity as the line at infinity.

the main problem about this hypothesis is that our procedure mentioned in this paper can be done to the x axis but not sure to the y axis which imply some distinction between real numbers and imaginary numbers in the meaningful of residues.

problem 18:

a new algorithm to statistic the number of mails we send and receive basing on java?

This example introduce an algorithm which can statistic the number of

sending and receiving mails.We add interception to the button first. If we send

mail ,it will record .By the try() and catch() method, we can calcular the

numbers of mails. Then we use judgement module，through archieving files，

we can judge whether it is sending or receiving mails by comparing the

filenames .Finally，we use database module to refresh timely, making the array

contrast personal information again, do query and recalculate from zero ,which

produce new calculations.

url: vixra.org/abs/1302.0057 (page 11-15)