Graduate Study in Applied Mathematics and Statistics at NJIT Why go to graduate school? Why study applied mathematics? What is applied mathematics? Graduate

Graduate Study in Applied Mathematics and Statistics at NJIT

Graduate Study in Applied Mathematics and Statistics at NJIT

Why go to graduate school?

Why study applied mathematics?What is applied mathematics?

Graduate study at NJITWhat do we do in the Department of Mathematical Sciences?

Why go to graduate school?Why go to graduate school?

A MS or a PhD degree is necessary for most advanced careers in today’s job market.

Engineering, R&DEducationFinancial industryPharmaceutical companies

Graduate education develops skills that are useful in a number of professions.

Why study applied mathematics?Why study applied mathematics?

Applied mathematics bridges disciplines: math, physics, engineering, biology, computer science, and others.

Education in applied mathematics includes exposure to various mathematical techniques, develops precise ways of thinking, and an ability to find creative solutions to real-life problems.

What do applied mathematicians do?What do applied mathematicians do?

They solve problems!What problems? From rocket science to development of new drugs to modeling the stock market and many others.

Where? In industry, government laboratories, and universities.

Why? Because all of these fields require analysis and well-trained mathematicians to stay at the leading edge

What is taught in a graduate program in applied mathematics?

What is taught in a graduate program in applied mathematics?

A variety of mathematical methods: modeling, analysis, asymptotics, and scientific computing.

Graduate students are exposed to other disciplines (physics, biology, chemistry, engineering) and are therefore able to apply their knowledge widely.

Training in applied mathematics allows for flexibility in professional life after graduate school.

Why study Applied Mathematics at the Department of Mathematical Sciences at NJIT?

Why study Applied Mathematics at the Department of Mathematical Sciences at NJIT?

The Department is ranked among TOP 10 Mathematics Department in the US, as reported by the Chronicle of Higher Education.

Work with world class faculty performing cutting edge research funded by NSF, NIH, AFOSR, DOE, ONR, HHMI, etc.

We provide personal attention to our students and ensure that they have all the necessary support to succeed in today’s demanding job market.

What do we do?What do we do?

We span of wide variety of research areas:

Applied Probability and Statistics

Mathematical Biology

Fluid Dynamics

Materials Science

Wave Propagation

What will a PhD student find at NJIT?What will a PhD student find at NJIT?

Nationally and internationally recognized faculty involved in projects that are at the forefront of current research.

State-of-the-art computing facilities, including large clusters for parallel computing.

Full support (fellowship, tuition) is available for qualified candidates.

Assistance with accommodation, relocation, etc.

Office, computer and other facilities.

What are PhD studies like?What are PhD studies like?

PhD candidates take a set of core courses taught by faculty who are committed to and enjoy graduate education.

After completing core course work, a student chooses a specialization and the most appropriate PhD advisor.

This process is helped by faculty-student seminars and daily interaction between students and faculty.

What is our track record?What is our track record?

Our past students have gone on to:

Academia, both research and teaching oriented institutions

Government Laboratories

High-tech startup companies

Pharmaceutical companies

The financial industry

List of currently supported research projects in the Department of Mathematical Sciences

List of currently supported research projects in the Department of Mathematical Sciences

Department of Mathematical Sciences research projects

Department of Mathematical Sciences research projects

Equipment and Educational GrantsEquipment and Educational Grants

Major Research Instrumentation – Beowulf Parallel Cluster

This 64-node computer cluster includes a total of 256 GB of memory, mass storage devices, scientific software, and hardware for a high speed Myrinet network. The machine is dedicated to the support of research by faculty and graduate students in the Department of Mathematical Sciences and the Center for Applied Mathematics and Statistics.

Computation and Communication: Promoting Research Integration in Science and Mathematics (C2PRISM)

This grant, supported by NSF's GK-12 program, combines PhD level research with high school teaching to infuse computation in the math and science curricula of urban schools in Newark, New Jersey. Graduate student Fellows will obtain hands-on instructional experience and improve communication skills, teachers will learn and experience current math and science research methods and improve computation skills, and high school students will benefit from having scientific role models and will learn computational skills in an applied context.

Equipment and Educational GrantsEquipment and Educational Grants

Mathematical BiologyMathematical Biology

Effects of Short-term Synaptic Plasticity in Feed-back Neuronal Networks

The property of a synapse to change its strength as a function of frequency is known as short-term synaptic plasticity. This project investigates the role of this property in the context of feed-back neuronal networks arising in Central Pattern Generating Network. It is conjectured that plasticity plays a large role in expanding the number of possible stable rhythmic states in such networks as well as the robustness of these states to perturbations.

Role of Neuromodulators and Activity in the Regulation of Ionic Currents and Neuronal Network Activity

The generation of rhythms in the nervous system is crucial to the survival of animals since they are involved in the production of vital functions (e.g. heart beat, respiration, locomotion, etc.) and the generation of cognitive functions (e.g. memory, awareness, sleep/wake cycles, etc). This project examines the mechanisms by which neuromodulators and neuronal networks' own activity regulate rhythmic patterns generation in a simple model system.


Gap Junction Role in Network Function

Gap junctions have been shown to affect generation of oscillations and synchronization of rhythmic activity between neurons, underlie coincidence detection and constitute the paths of intercellular exchange for biologically active molecules. It is proposed to apply a complementary approach using electrophysiology, modeling, mathematical analysis and cell biology to address the role gap junction position along the dendritic tree in network function, the effect gap junction coupling on the measurements of voltage-dependent ionic conductances, and the role of dendrite diameter in sustaining synchronized fast and slow oscillations in gap-junctionally coupled networks.

Presynaptic Calcium Dynamics, Calcium Buffers, and The Mechanisms of Synaptic Facilitation.

This project is focused on the modeling of cell calcium dynamics responsible for neurotransmitter release at chemical synapses, and the transient changes in synaptic transmission strength. It involves 3D simulations of calcium influx into the cell through calcium channels, its diffusion inside the cytosol, and the interaction of calcium ions with calcium-binding molecules called calcium buffers.


Analysis of Spatiotemporal Signal Processing in Developmental Patterning

This project is concerned with developing analytical approaches for studying the spatiotemporal dynamics in patterning networks, making extensive use of the techniques of calculus of variations. The research is closely linked to the experimental and modeling work on pattern formation in Drosophila development.

Cortical Processing Across Multiple Time and Space-scales

The primary visual cortex (V1) is a complex integrated circuit that performs fundamental tasks in processing of visual information by the brain. In this complex system, the dynamics is induced and influenced by network components at many scales from fast, short-range interactions to large cortical areas that encompass many modules (e.g., the so-called orientation hypercolumns) with long-range connections. To model the emergent biological functions, coarse-grained effective representations for evolving large-scale network dynamics that exploit the modular nature of the network has been developed.

Fluid DynamicsFluid Dynamics

Analysis and Numerical Computations of Free Boundaries in Fluid Dynamics: Surfactant Solubility and Elastic Fibers

This project studies the influence of surfactants on the behavior of interfacial flows, and on the interaction of elastic fibers with a free fluid interface. A particular focus is on the development of hybrid analytical-numerical methods that are relevant to effects of surfactant solubility and fiber elasticity.

Propagation of Fronts in Porous Media Combustion

The project is focused on mathematical analysis of propagation and stability of reactive fronts in inert porous media. Potential applications arise in micro-scale combustion, such as energy production and micro-propulsion.


Bridging the Spatial and Temporal Scales in Dense Granular Systems

This project centers on multiscale modeling of information propagation through dense granular matter. Relevant application range from the fields such as oil recovery and vibrofluidized beds to some humanitarian efforts, such as detection of land mines.

Thermal Effects on the Dynamics of Singularity Formation in Viscous Threads

A class of free boundary problems governed by nonlinear partial differential equations is studied where localized heating controls the dynamics of the formation of singularities in surface tension driven flows. There are fundamental issues on how temperature effects on viscosity and surface tension play a major role in shaping viscous threads.


Interaction between flow and topography in Interfacial Electrohydrodynamics

This project is concerned with modeling and simulations of electrified film flows over topography. Applications arise in the manufacture of micro-electronics and in micro- and nano-fluidic based processes such as mixing enhancement.

Numerics and Analysis of Singularities for the Euler Equations

This project involves the development of methods for finding singular solutions to partial differential equations, with applications to the Euler equations and related problems. The investigator's approach to constructing singular solutions is by complementary analytical and numerical methods, and will build on their previous results involving the numerical construction of complex, singular Euler solutions.

Highly Nonlinear Wave Phenomena in The Ocean

This research focuses on the development of simple but reliable mathematical and numerical models to describe the evolution of nonlinear surface and internal gravity waves and is expected to enhance our prediction capability of extreme waves in the ocean.

Wave PropagationWave Propagation

Mathematical Methods for Wave Interactions

This is concerned with understanding the complicated interactions that arise between waves when they collide. Simple systems can display chaotic scattering, where the speed with which a wave exits a collision depends in a complicated way on the speed with which it enters. New mathematical descriptions of this process, that allow the application of mathematical techniques from dynamical systems, are being developed.

Processing of Ceramic Materials by Microwave and Ohmic Heating

The rapid heating of ceramics by microwave and other forms of electrical energy is the basis of several emerging industrial processes such as sintering, fiber coating, joining, and chemical vapor infiltration. This project investigates heating of thin ceramic samples in a highly tuned microwave cavity, microwave assisted chemical vapor infusion, novel processes using microwaves to selectively heat ceramic fibers, and electric discharge densification.

Wave PropagationWave Propagation

Patterns, Stability, and Thermal Effects in Parametric Gain Devices

This project studies the impact of heating caused by the absorption of optical energy on the operation of frequency conversion devices based on parametric gain. These laser-like devices are used for purposes ranging from the jamming of heat-seeking missiles to the spectroscopic analysis of gaseous samples.

Simulation of Rare Events in Lightwave Systems

This project aims to develop new analytical and numerical tools for the analysis of rare events (specifically, transmission failures) in fiber-based optical communications. These methods are intended to be applicable to a variety of settings involving stochastically driven nonlinear wave equations.

Nonlinear DynamicsNonlinear Dynamics

Accuracy and Stability of Computational Representations of Swept Volume Operations

This project is concerned with the development of theoretical and practical means for algorithmically representing and rendering complex geometric objects - especially those created by sweeping geometric objects through space - in a topologically and geometrically consistent manner. The main tools used come from nonlinear dynamics, and the new field of computational topology. Applications arise in computer aided design and manufacturing, computer graphics, and computational approximation of physical phenomena.

Applied StatisticsApplied Statistics

Analysis of Volatile Organic Compounds (VOCs) in the Vicinity of the Teterboro Airport

This project is focused on the estimation of VOC concentration in eight municipalities in the vicinity of the Teterboro airport as a function of several environmental variables, the frequency of landings/take-offs, and other factors such as the distance from the airport and the time spent indoors and outdoors.

Wetland Monitoring and Assessment Using Hyperspectral Remote Sensing

This project is concerned with the analysis of the data collected from experiments designed to examine the hypothesis that hyperspectral images of wetland vegetation alone can be used as a surrogate for mapping wetland relative elevation and sediment biogeochemistry conditions.

Applied StatisticsApplied Statistics

Volume Testing of Electronic Voting Machines with Voter-Verified Paper Record System

This project is focused on the testing the electronic voting machines with voter-verified paper record systems by conducting volume testing to simulate the actual voting during general elections to examine whether or not these machines meet the requirements. Fractional factorial designs are used to select 12 voting scenarios to represent millions of possible combinations pf voting patterns.

BiostatisticsBiostatistics

Tests for Establishing Noninferiority in Clinical Trials

In this project improved methods to test data based on primary efficacy variables to evaluate new drug for noninferiority are developed. Statistical method of bootstrapping is used to achieve these results and shown to have improvement over the conventional methods. In such research, test drugs would have already shown statistical and clinical meaningfulness when compared to placebo.

Adaptive Two-Stage Design with a Control Group

In this research, a 2-stage design, including a control is considered. The design considered here is a modification of Simon's two-stage design. This design is generally used to improve quality of life of cancer patients and to reduce unnecessary risk of their being exposed to inferior treatment. SAS program is used to calculate optimal solutions for the sample size as well as the critical values at both Stages 1 and 2 of the design for given false positive and false negative error rates.

Examples of current projects in the department…

Examples of current projects in the department…

Department of Mathematical Sciences Current projects

Department of Mathematical Sciences Current projects

The Fountain MapThe Fountain MapFor students who are “thirsty for mathematics” we bring

you…

The fountain map is an example of a general construction of new models for chaotic dynamics.

The fountain map is a discrete dynamical The fountain map is a discrete dynamical system that exhibits complicated dynamics system that exhibits complicated dynamics when iterated, i.e. applied over and over again.when iterated, i.e. applied over and over again.

The fountain map is a discrete dynamical The fountain map is a discrete dynamical system that exhibits complicated dynamics system that exhibits complicated dynamics when iterated, i.e. applied over and over again.when iterated, i.e. applied over and over again.

This chaotic system is novel in the sense that it makes use of heteroclinic dynamics. As the parameters are varied, the unstable manifolds interact with the stable manifolds (invariant curves) to produce interesting iterate graphs like the one to the right.

The The Map: Map:The The Map: Map:

The fountain map exhibits what are called Hopf bifurcations of very large periods. Below is a case when the period is 19.

Initial point= (0.1,0.1)

Parameters:

For a dense set of points

Click white screen TWICE to view movie Click screen TWICE to view movie

Rotating the planar map about the z-axis yields higher dimensional versions of the Fountain Map.

The rotation construction can be extended to N dimensions.

Notice that the planar map is being rotated about the z axis. This rotation construction traces out a torus (doughnut) in three-dimensional space. Here two points (green and blue) are being iterated

Click white screen TWICE to view movie

You can iterate the fountain map presentation by watching it again.

Chaotic Scattering of Solitary Waves

Chaotic Scattering of Solitary Waves

Department of Mathematical SciencesDepartment of Mathematical Sciences

Solitary WavesSolitary Waves

An interesting question is “What happens when two such waves collide?”

Many systems from mathematical physics support solitary wave solutions – waves with a definite shape moving at a constant speed. These describe many phenomena including water waves and light propagation in optical fibers. Two types are “kinks” and “pulses.”

• One collision, then escape

• Capture following the collision

• Two collisions before escape

The only difference among the 3 movies is the initial speed!

The φ4 equation (which arises in everything from galaxy formation to magnetism) supports “kink”-type solitary waves. Let’s watch some kink-antikink collisions:

Dependence of Escape Velocity on Initial Velocity

Dependence of Escape Velocity on Initial Velocity

This is an example of a phenomenon called chaotic scattering: the speed with which they separate depends in a very complicated way on their initial speed.

• Color indicates number of collisions before escape

Reducing the Model (Part I)Reducing the Model (Part I)

Previous researchers have shown the behavior can be well approximated by the solution to a system of ODE’s which describes the evolution of just two parameters over time:

d 2 X

dt2

d 2U ( X )

dt2

dF( X )

dt0,

d 2a

dt2 a F( X ) 0

The equation is a partial differential equation (PDE). It describes the evolution of a shape over time:

(x) 4

tt xx 3 0

Reducing the Model (Part II)Reducing the Model (Part II)

d 2 X

dt2

d 2U ( X )

dt2 dF( X )

dt0,

d 2a

dt2 a F( X ) 0

Here X(t) is the distance between two waves, a(t) is the measures the change in to the waves’ shapes, and U(X) and F(X) are coupling functions that decay rapidly away from X = 0.

Using dynamical systems theory, we derived an iterated map approximation--a much simpler system

Here tn is time of nth collision, and En is the stored energy after n collisions. Before collision E0 > 0. After one collision E1 < 0, and collisions continue until En > 0, followed by escape. The final velocity of escape is a function of En

En1

En t

1,t

2,...,t

n t

n1t

nT E

n

Results with Reduced ModelsResults with Reduced Models

Results with Reduced ModelsResults with Reduced Models

Variation of “number of bounces” with ε and initial velocity. The vertical (ε) axis is the strength of interaction of modes, and the horizontal (v) axis is the velocity. A black curve shows dependence of critical speed on ε. To the right of the black curve the waves separate after one collision, to the left the numbers of collisions prior to separation are color-coded. These structures – with similar details at all scales – are fractals, a beautiful discovery of modern mathematics.

Chaotic Microwave Heating & Processing of Ceramics



Alumina (Al2O3) is a typical refractory ceramic. You have seen it as the white cylindrical insulator near the electrodes of a spark plug in an automobile engine.



Other examples occur in:

Rocket motor nozzlesJet engine turbine blades

Regenerative particle filtersHeat exchanger and chemical reactor tubes

chemical vapor injection, and combustion synthesis.

Useful processes are:

Sintering, joining (or welding),

Advantages of microwave heat processing

Advantages of microwave heat processing

Rapid internal heating of material: microwaves penetrate the material directly, so no reliance on diffusion of heat.

Non-uniform (localized) heating.

An advantage or a disadvantage: hot-spot formation, and very high localized temperature can cause thermal runaway (i.e., meltdown).

Process control is important.

Applied MathematicsApplied Mathematics

Direct computer simulation of the heat equation and Maxwell’s equations takes ~ 24-48 hours cpu time.

Modeling uses disparate space and time scales:from 1 mm to 10 cm, and 0.25 10-9 s to 1 min to 10 min.

More important, it leads to simpler equations that can be solved in ~2 min cpu time, plus understanding of fundamental physical processes and experimental observations.

A Lab ExperimentA Lab Experiment

Microwave welding of mullite (3Al2O3+2SiO2) at about 13000C (courtesy of DHR Inc.)

Model predictions for temperature distribution of a

microwave heated slab or plate Model predictions for temperature distribution of a

microwave heated slab or plate

Hot Spot on a Ceramic Slab

Hot Stripe on a Ceramic Slab

Hemoglobin and Sickle Cell Anemia Hemoglobin and Sickle Cell Anemia

Hemoglobin is an iron-containing, oxygen-transport metalloprotein in the red blood cells of vertebrates and other animals.

Mutations in the genes for the hemoglobin protein result in a group of hereditary diseases. The best known is sickle cell anemia.

It is caused by a point mutation in the β-globin chain of hemoglobin, replacing the amino acid glutamic acid with the less polar amino acid valine at the sixth position of the β chain (hemoglobin S or HbS).

This was the first time a genetic disease was linked to a mutation of a specific protein.

From change at molecular level to macroscopic mechanical From change at molecular level to macroscopic mechanical property change:property change:

From change at molecular level to macroscopic mechanical From change at molecular level to macroscopic mechanical property change:property change:

Sickle Cell: hydrophobic

Normal: hydrophilic

Interactive ModelInteractive Model

Single Protein ModelSingle Protein Model

After One Nano SecondAfter One Nano Second

Stochastic Finite Element Modeling of Cell Cytoskeleton: Stochastic Finite Element Modeling of Cell Cytoskeleton: A Fluid-Solid SystemA Fluid-Solid System

Normal and Sickle RBCs Flow in Normal and Sickle RBCs Flow in CapillaryCapillary

The Role of Dendrite Diameter in Determining Activity in Electrically-coupled

Neuronal Networks

The Role of Dendrite Diameter in Determining Activity in Electrically-coupled

Neuronal Networks


1 2

Rc

Gap junctions connect the inside of cells and carry electrical currents between neurons directly by ion flow.

Gap junctions can be represented electrically as a resistor, and the current they carry is given by Ohm’s law:

I1-2 = Rc(V1-V2)

But this mathematical simplicity is deceiving.

Cell to Cell Communication via “Gap Junctions”

Cell to Cell Communication via “Gap Junctions”

Neurons Can Form Networks of Different Sizes by Connecting Only Via Gap Junctions

Neurons Can Form Networks of Different Sizes by Connecting Only Via Gap Junctions

• Voltage signals can flow between cells in small or large networks (red arrows below), and complex behaviors can emerge, such as voltage oscillations.

• Also, we can study the effects of the diameter of the dendrites on network activity because V = f(d) (voltage along cables depend of diameter)

• Voltage signals can flow between cells in small or large networks (red arrows below), and complex behaviors can emerge, such as voltage oscillations.

• Also, we can study the effects of the diameter of the dendrites on network activity because V = f(d) (voltage along cables depend of diameter).

In a complex network, activity will spread, a path for the spread of activity is selected by the dendrite diameter d, and a recurrent wave of

activity through the network will occur

In a complex network, activity will spread, a path for the spread of activity is selected by the dendrite diameter d, and a recurrent wave of

activity through the network will occur

d = 3.5 μmThe Network:

Each dot is a neuron with 3 dendrites as before.

Neurons are connected to 1, 2 or 3 others by gap junctions between dendrites

We start by stimulating cell 1 (click to show wave):

Cells activate in a sequence,

and all cells in the network activate periodically

d = 2 μm

A different path is selected at a different diameter. The speed of the wave is also different.

A different path is selected at a different diameter. The speed of the wave is also different.

Exactly the same network,only different dendrite diameter We start by

stimulating cell 1

(click to show wave):

A new spreading pattern appears, with different sequence

and also at a new frequency of periodic activity

Signal propagation in particulate systems

Signal propagation in particulate systems

Idea: to understand how information propagates in a large number of particles (sand, soil…)

Technique: discrete element (i.e., molecular dynamics) simulations modeling the details of particles’ interaction

Computing: large scale simulations of hundreds of thousands of particles

Signal Propagation Example: Time Evolution of the Forces Between Granulars

Signal Propagation Example: Time Evolution of the Forces Between Granulars

(click to play)

Signal Propagation Challenge: Understand The Propagating Waves

Signal Propagation Challenge: Understand The Propagating Waves

the movie shows the elastic

(compressional) energy of the

particles

Modified Simon’s Two-Stage Design with a Control GroupModified Simon’s Two-Stage Design with a Control Group


SIMON’S TWO-STAGE DESIGNSIMON’S TWO-STAGE DESIGN

Simon’s 2-stage design is a single-group design for a binary response, which is commonly utilized in Phase II clinical trials, especially in the oncology area (Simon, 1989).

It tests the null hypothesis that the response rate to treatment, p, is at most equal to p0, i.e., treatment is not effective, against the alternative that p is at least p1 i.e., the treatment is effective.

If fewer than k1 responders are observed in Stage 1, the trial is terminated and the null hypothesis is not rejected.

If at least k1 responders are observed, an additional n2

subjects are enrolled in Stage 2. At the completion of Stage 2, if fewer than k (> k1 ) responders are observed, the null hypothesis is not rejected; otherwise, alternative hypothesis is accepted.

Advantage of Simon’s two-stage design is that it reduces unnecessary risk of exposing subjects to inferior treatment. However, without a control group, results could be unreliable, especially when p0 and p1 values are unrealistic.

Given p0 , p1, α, β, an optimal design is determined by enumeration using exact binomial probabilities.

SIMON’S TWO-STAGE DESIGNSIMON’S TWO-STAGE DESIGN

MODIFIED SIMON’S TWO-STAGE DESIGN WITH CONTROL GROUP


Let pi denote the true response rate for treatment i.

Large values of p1-p2 indicates superiority of treatment 1 over treatment 2.

We are interested in a two-stage design to test H0: p1-p2 ≤ Δ0 (i.e., treatment 1 is not superior to treatment 2) versus Ha: p1-p2 ≥ Δ1 (i.e., treatment 1 is superior to treatment 2).

In Stage 1, n1 patients per group are randomized into the two groups. H0 is not rejected if the difference between the number of responders under treatment 1 and treatment 2 is smaller than k1. If H0 is not rejected in Stage 1 the trial is terminated.

However, if the difference between the number of responders under treatment 1 and treatment 2 is at least k1, then n2 additional patients per group are randomized into the two groups in Stage 2.

If the difference between the number of responders under treatment 1 and treatment 2 is less than k, when the sample sizes per group are n1+ n2, again H0 is not rejected.



OPTIMAL DESIGNOPTIMAL DESIGNn = 15, max alpha = 0.05, min power = 0.70, p1 = 0.45, p2 = 0.10.

Sol# n1 n2 k1 k Alpha PowerP_Reject

Ho1_error

P_Reject

Ho1_power

P_Reject

Ha1_c

P_Reject

Ha1_a

1 10 5 1 4 0.0176 0.7899 0.0051 0.5048 0.6563 0.05132 10 5 2 4 0.0164 0.7655 0.0051 0.5048 0.8782 0.13693 11 4 1 4 0.0179 0.7939 0.0069 0.5766 0.6481 0.04134 11 4 2 4 0.0174 0.7820 0.0069 0.5766 0.8665 0.11065 11 4 3 4 0.0139 0.7215 0.0069 0.5766 0.9642 0.23906 12 3 1 4 0.0180 0.7956 0.0090 0.6400 0.6411 0.03347 12 3 2 4 0.0179 0.7919 0.0090 0.6400 0.8557 0.08968 12 3 3 4 0.0158 0.7562 0.0090 0.6400 0.9580 0.19729 13 2 2 4 0.0179 0.7955 0.0113 0.6952 0.8456 0.0728

10 13 2 3 4 0.0171 0.7808 0.0113 0.6952 0.9518 0.162611 14 1 3 4 0.0174 0.7927 0.0138 0.7428 0.9455 0.1340

For more informationFor more information

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For application info see:http://www.njit.edu/graduatestudies/

Contact us by email at: [email protected] or – see the list of faculty at

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Graduate Study in Applied Mathematics and Statistics at NJIT Why go to graduate school? Why study applied mathematics? What is applied mathematics? Graduate